The principles of Leibniz's logic and, arguably, of his whole philosophy, reduce to two:. The formal logic that emerged early in the 20th century also requires, at minimum, unary negation and quantified variables ranging over some universe of discourse. Leibniz published nothing on formal logic in his lifetime; most of what he wrote on the subject consists of working drafts.
In his book History of Western Philosophy , Bertrand Russell went so far as to claim that Leibniz had developed logic in his unpublished writings to a level which was reached only years later. Russell's principal work on Leibniz found that many of Leibniz's most startling philosophical ideas and claims e. He regarded such relations as real qualities of things Leibniz admitted unary predicates only : For him "Mary is the mother of John" describes separate qualities of Mary and of John.
Notably, Leibniz also declared space and time to be inherently relational. Although the mathematical notion of function was implicit in trigonometric and logarithmic tables, which existed in his day, Leibniz was the first, in and , to employ it explicitly, to denote any of several geometric concepts derived from a curve, such as abscissa , ordinate , tangent , chord , and the perpendicular.
Leibniz also believed that the sum of an infinite number of zeros would equal to one half using the analogy of the creation of the world from nothing. Leibniz's discoveries of Boolean algebra and of symbolic logic , also relevant to mathematics, are discussed in the preceding section. The best overview of Leibniz's writings on calculus may be found in Bos Leibniz arranged the coefficients of a system of linear equations into an array, now called a matrix , in order to find a solution to the system if it existed.
Leibniz laid down the foundations and theory of determinants , although Seki Kowa discovered determinants well before Leibniz. Finding the determinant of a matrix using this method proves impractical with large n , requiring to calculate n!
This method for solving systems of linear equations based on determinants was found in by Leibniz Cramer published his findings in Leibniz wrote that circles "can most simply be expressed by this series, that is, the aggregate of fractions alternately added and subtracted. Leibniz is credited, along with Sir Isaac Newton , with the discovery of calculus differential and integral calculus. Leibniz did not publish anything about his calculus until The concept became more transparent as developed through Leibniz's formalism and new notation.
In addition, the theorem that tells how and when to differentiate under the integral sign is called the Leibniz integral rule. Leibniz exploited infinitesimals in developing calculus, manipulating them in ways suggesting that they had paradoxical algebraic properties.
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A recent study argues that Leibnizian calculus was free of contradictions, and was better grounded than Berkeley's empiricist criticisms. From until his death, Leibniz was engaged in a dispute with John Keill, Newton and others, over whether Leibniz had invented calculus independently of Newton.
This subject is treated at length in the article Leibniz—Newton calculus controversy. The use of infinitesimals in mathematics was frowned upon by followers of Karl Weierstrass ,  [ citation needed ] but survived in science and engineering, and even in rigorous mathematics, via the fundamental computational device known as the differential.
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Beginning in , Abraham Robinson worked out a rigorous foundation for Leibniz's infinitesimals, using model theory , in the context of a field of hyperreal numbers. The resulting non-standard analysis can be seen as a belated vindication of Leibniz's mathematical reasoning.
Robinson's transfer principle is a mathematical implementation of Leibniz's heuristic law of continuity , while the standard part function implements the Leibnizian transcendental law of homogeneity. Leibniz was the first to use the term analysis situs ,  later used in the 19th century to refer to what is now known as topology.
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There are two takes on this situation. On the one hand, Mates, citing a paper in German by Jacob Freudenthal , argues:. He mistakenly credits Leibniz with originating this concept. But Hideaki Hirano argues differently, quoting Mandelbrot : . To sample Leibniz' scientific works is a sobering experience. Next to calculus, and to other thoughts that have been carried out to completion, the number and variety of premonitory thrusts is overwhelming. We saw examples in "packing", My Leibniz mania is further reinforced by finding that for one moment its hero attached importance to geometric scaling.
In Euclidis Prota The straight line is a curve, any part of which is similar to the whole, and it alone has this property, not only among curves but among sets.
Thus the fractal geometry promoted by Mandelbrot drew on Leibniz's notions of self-similarity and the principle of continuity: Natura non facit saltus. As for "packing", Leibniz told his friend and correspondent Des Bosses to imagine a circle, then to inscribe within it three congruent circles with maximum radius; the latter smaller circles could be filled with three even smaller circles by the same procedure. This process can be continued infinitely, from which arises a good idea of self-similarity.
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Leibniz's improvement of Euclid's axiom contains the same concept. 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 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 held a relationist notion of space and time, against Newton's substantivalist views. Leibniz's relationism, on the other hand, describes space and time as systems of relations that exist between objects. The rise of general relativity and subsequent work in the history of physics has put Leibniz's stance in a more favorable light.
One of Leibniz's projects was to recast Newton's theory as a vortex theory.
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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. In addition to his theories about the nature of reality, Leibniz's contributions to the development of calculus have also had a major impact on physics. Leibniz's vis viva Latin for "living force" is m v 2 , 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. 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. 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. Psychology had been a central interest of Leibniz. His discussions in the New Essays and Monadology often rely on everyday observations such as the behaviour of a dog or the noise of the sea, and he develops intuitive analogies the synchronous running of clocks or the balance spring of a clock.
Bodies act according to the laws of efficient causes, i. And these two realms, that of efficient causes and that of final causes, harmonize with one another. Leibniz found his most important interpreter in Wilhelm Wundt , founder of psychology as a discipline. 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.
His theory regarding consciousness in relation to the principle of continuity can be seen as an early theory regarding the stages of sleep. In this way, Leibniz's theory of perception can be viewed as one of many theories leading up to the idea of the unconscious. 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.