Leibniz's writings are currently discussed, not only for their anticipations and possible discoveries not yet recognized, but as ways of advancing present knowledge. Much of his writing on physics is included in Gerhardt's Mathematical Writings.
Leibniz contributed a fair amount to the statics and dynamics emerging around him, often disagreeing with Descartes and Newton. He devised a new theory of motion (dynamics) based on kinetic energy and potential energy, which posited space as relative, whereas Newton was thoroughly convinced that space was absolute. An important example of Leibniz's mature physical thinking is his Specimen Dynamicum of 1695.
Until the discovery of subatomic particles and the quantum mechanics governing them, many of Leibniz's speculative ideas about aspects of nature not reducible to statics and dynamics made little sense. For instance, he anticipated Albert Einstein by arguing, against Newton, that space, time and motion are relative, not absolute: "As for my own opinion, I have said more than once, that I hold space to be something merely relative, as time is, that I hold it to be an order of coexistences, as time is an order of successions." Leibniz's rule for the derivatives of products is an important, if often overlooked, step in many proofs in diverse fields of physics. The principle of sufficient reason has been invoked in recent cosmology, and his identity of indiscernibles in quantum mechanics, a field some even credit him with having anticipated in some sense. Those who advocate digital philosophy, a recent direction in cosmology, claim Leibniz as a precursor.
The vis viva
Leibniz's vis viva (Latin for living force) is mv2, twice the modern kinetic energy. He realized that the total energy would be conserved in certain mechanical systems, so he considered it an innate motive characteristic of matter. Here too his thinking gave rise to another regrettable nationalistic dispute. His vis viva was seen as rivaling the conservation of momentum championed by Newton in England and by Descartes in France; hence academics in those countries tended to neglect Leibniz's idea. In reality, both energy and momentum are conserved, so the two approaches are equally valid.
Other natural science
By proposing that the earth has a molten core, he anticipated modern geology. In embryology, he was a preformationist, but also proposed that organisms are the outcome of a combination of an infinite number of possible microstructures and of their powers. In the life sciences and paleontology, he revealed an amazing transformist intuition, fueled by his study of comparative anatomy and fossils. One of his principal works on this subject, Protogaea, unpublished in his lifetime, has recently been published in English for the first time. He worked out a primal organismic theory. In medicine, he exhorted the physicians of his time—with some results—to ground their theories in detailed comparative observations and verified experiments, and to distinguish firmly scientific and metaphysical points of view.
Much of Leibniz's work went on to have a great impact on the field of psychology. His theory regarding consciousness in relation to the principle of continuity can be seen as an early theory regarding the stages of sleep. He believed that by the principle that phenomena found in nature were continuous by default, it was likely that the transition between conscious and unconscious states had intermediary steps. Though Leibniz's ideas regarding pre-established harmony were rejected by many, psychologists embraced his ideas of psychophysical parallelism. This idea refers to the mind–body problem, stating that the mind and brain do not act upon each other, but act alongside each other separately but in harmony.
Leibniz believed that the mind had a very active role in perception, and plays a much larger role in sensory input. He focused heavily on perception, distinguishing between the type of perception where we are conscious of a stimulus, and the other which is being aware of a distinct perception. He thought that there are many petites perceptions, or small perceptions of which we perceive but of which we are unaware. For example, when a bag of rice is spilled, we see the rice but are not necessarily aware of how many grains are in the pile. With this principle, there are an infinite number of perceptions within us at any given time of which we are unaware. For this to be true, there must also be a portion of the mind of which we are unaware at any given time. In this way, Leibniz's theory of perception can be viewed as one of many theories leading up to the idea of the unconscious. Additionally, the idea of subliminal stimuli can be traced back to his theory of small perceptions. Leibniz was a direct influence on Ernst Platner, who is credited with originally coining the term Unbewußtseyn (unconscious).
Leibniz's ideas regarding music and tonal perception went on to influence the laboratory studies of Wilhelm Wundt.
In public health, he advocated establishing a medical administrative authority, with powers over epidemiology and veterinary medicine. He worked to set up a coherent medical training program, oriented towards public health and preventive measures. In economic policy, he proposed tax reforms and a national insurance program, and discussed the balance of trade. He even proposed something akin to what much later emerged as game theory. In sociology he laid the ground for communication theory.
In 1906, Garland published a volume of Leibniz's writings bearing on his many practical inventions and engineering work. To date, few of these writings have been translated into English. Nevertheless, it is well understood that Leibniz was a serious inventor, engineer, and applied scientist, with great respect for practical life. Following the motto theoria cum praxi, he urged that theory be combined with practical application, and thus has been claimed as the father of applied science. He designed wind-driven propellers and water pumps, mining machines to extract ore, hydraulic presses, lamps, submarines, clocks, etc. With Denis Papin, he invented a steam engine. He even proposed a method for desalinating water. From 1680 to 1685, he struggled to overcome the chronic flooding that afflicted the ducal silver mines in the Harz Mountains, but did not succeed.
Leibniz may have been the first computer scientist and information theorist. Early in life, he documented the binary numeral system (base 2), then revisited that system throughout his career. He anticipated Lagrangian interpolation and algorithmic information theory. His calculus ratiocinator anticipated aspects of the universal Turing machine. In 1934, Norbert Wiener claimed to have found in Leibniz's writings a mention of the concept of feedback, central to Wiener's later cybernetic theory.
In 1671, Leibniz began to invent a machine that could execute all four arithmetical operations, gradually improving it over a number of years. This "Stepped Reckoner" attracted fair attention and was the basis of his election to the Royal Society in 1673. A number of such machines were made during his years in Hanover, by a craftsman working under Leibniz's supervision. It was not an unambiguous success because it did not fully mechanize the operation of carrying. Couturat reported finding an unpublished note by Leibniz, dated 1674, describing a machine capable of performing some algebraic operations. Leibniz also devised a (now reproduced) cipher machine, recovered by Nicholas Rescher in 2010. In 1693, Leibniz released to the public a design of a machine which could, in theory, integrate differential equations.
Leibniz was groping towards hardware and software concepts worked out much later by Charles Babbage and Ada Lovelace. In 1679, while mulling over his binary arithmetic, Leibniz imagined a machine in which binary numbers were represented by marbles, governed by a rudimentary sort of punched cards. Modern electronic digital computers replace Leibniz's marbles moving by gravity with shift registers, voltage gradients, and pulses of electrons, but otherwise they run roughly as Leibniz envisioned in 1679.
While serving as librarian of the ducal libraries in Hanover and Wolfenbuettel, Leibniz effectively became one of the founders of library science. The latter library was enormous for its day, as it contained more than 100,000 volumes, and Leibniz helped design a new building for it, believed to be the first building explicitly designed to be a library. He also designed a book indexing system in ignorance of the only other such system then extant, that of the Bodleian Library at Oxford University. He also called on publishers to distribute abstracts of all new titles they produced each year, in a standard form that would facilitate indexing. He hoped that this abstracting project would eventually include everything printed from his day back to Gutenberg. Neither proposal met with success at the time, but something like them became standard practice among English language publishers during the 20th century, under the aegis of the Library of Congress and the British Library.
He called for the creation of an empirical database as a way to further all sciences. His characteristica universalis, calculus ratiocinator, and a "community of minds"—intended, among other things, to bring political and religious unity to Europe—can be seen as distant unwitting anticipations of artificial languages (e.g., Esperanto and its rivals), symbolic logic, even the World Wide Web.
Advocate of scientific societies
Leibniz emphasized that research was a collaborative endeavor. Hence he warmly advocated the formation of national scientific societies along the lines of the British Royal Society and the French Academie Royale des Sciences. More specifically, in his correspondence and travels he urged the creation of such societies in Dresden, Saint Petersburg, Vienna, and Berlin. Only one such project came to fruition; in 1700, the Berlin Academy of Sciences was created. Leibniz drew up its first statutes, and served as its first President for the remainder of his life. That Academy evolved into the German Academy of Sciences, the publisher of the ongoing critical edition of his works.