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Gottfried Leibniz

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Gottfried Leibniz
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Gottfried Leibniz

Gottfried Wilhelm von Leibniz (also Leibnitz) (Leipzig July 1 (June 21 O.S.), 1646November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. Leibniz is credited with the term "function" (1694), which he used to describe a quantity related to a curve, such as a curve's slope or a specific point of said curve. Leibniz is generally, with Newton, jointly credited for the development of the modern calculus; in particular, for his development of the integral and the product rule.

Contents

Career and controversy

From A Short Account of the History of Mathematics (4th edition, 1908) by W. W. Rouse Ball (see Discussion)

His father died before he was six, and the teaching at the school to which he was then sent was inefficient, but his industry triumphed over all difficulties; by the time he was twelve he had taught himself to read Latin easily, and had begun Greek; and before he was twenty he had mastered the ordinary text-books on mathematics, philosophy, theology and law. Refused the degree of doctor of laws at Leipzig by those who were jealous of his youth and learning, Leibniz moved to Nuremberg. There, he wrote an essay on the study of law, that was dedicated to the Elector of Mainz, and led to his appointment by the elector on a commission for the revision of some statutes, from which he was subsequently promoted to the diplomatic service. In the latter capacity he supported (unsuccessfully) the claims of the German candidate for the crown of Poland. The violent seizure of various small places in Alsace in 1670 excited universal alarm in Germany as to the designs of Louis XIV; and Leibniz drew up a scheme by which it was proposed to offer German co-operation, if France liked to take Egypt, and use the possessions of that country as a basis for attack against Holland in Asia, provided France would agree to leave Germany undisturbed. This bears a curious resemblance to the similar plan by which Napoleon I proposed to attack England. In 1672 Leibniz went to Paris on the invitation of the French government to explain the details of the scheme, but nothing came of it.

At Paris he met Christiaan Huygens, who was then residing there, and their conversation led Leibniz to study geometry, which he described as opening a new world to him; though as a matter of fact he had previously written some tracts on various minor points in mathematics, the most important being a paper on combinations written in 1668, and a description of a new calculating machine. In January, 1673, he was sent on a political mission to London, where he stopped some months and made the acquaintance of Henry Oldenburg, John Collins, and others; it was at this time that he communicated the memoir to the Royal Society in which he was found to have been forestalled by Gabriel Mouton (http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Mouton.html).

In 1673 the Elector of Mainz died, and in the following year Leibniz entered the service of the Brunswick family; in 1676 he again visited London, and then moved to Hanover, where, till his death, he occupied the well-paid post of librarian in the ducal library. His pen was thenceforth employed in all the political matters which affected the Hanoverian family, and his services were recognized by honours and distinctions of various kinds, his memoranda on the various political, historical, and theological questions which concerned the dynasty during the forty years from 1673 to 1713 form a valuable contribution to the history of that time.

Leibniz's appointment in the Hanoverian service gave him more time for his favourite pursuits. He used to assert that as the first-fruit of his increased leisure, he invented the differential and integral calculus in 1674, but the earliest traces of the use of it in his extant notebooks do not occur till 1675, and it was not till 1677 that we find it developed into a consistent system; it was not published till 1684. Most of his mathematical papers were produced within the ten years from 1682 to 1692, and many of them in a journal, called the Acta Eruditorum (http://www.library.utoronto.ca/robarts/microtext/collection/pages/actaerud.html), founded by himself and Otto Mencke in 1682, which had a wide circulation on the continent.

Leibniz occupies at least as large a place in the history of philosophy as he does in the history of mathematics. Most of his philosophical writings were composed in the last twenty or twenty-five years of his life; and the points as to whether his views were original or whether they were appropriated from Spinoza, whom he visited in 1676, is still in question among philosophers, though the evidence seems to point to the originality of Leibniz. As to Leibniz's system on philosophy it will be enough to say that he regarded the ultimate elements of the universe as individual percipient beings whom he called monads. According to him the monads are centres of force, and substance is force, while space, matter, and motion are merely phenomenal; finally, the existence of God is inferred from the existing harmony among the monads. His services to literature were almost as considerable as those to philosophy; in particular, his overthrow of the then prevalent belief that Hebrew was the primeval language of the human race.

In 1700 the academy of Berlin was created on his advice, and he drew up the first body of statutes for it. On the accession in 1714 of his master, George I, to the throne of England, Leibniz was thrown aside as a useless tool; he was forbidden to come to England; and the last two years of his life were spent in neglect and dishonour. He died at Hanover in 1716. He was overfond of money and personal distinctions; was unscrupulous, as perhaps might be expected of a professional diplomatist of that time; but possessed singularly attractive manners, and all who once came under the charm of his personal presence remained sincerely attached to him. His mathematical reputation was largely augmented by the eminent position that he occupied in diplomacy, philosophy and literature; and the power thence derived was considerably increased by his influence in the management of the Acta Eruditorum.

The last years of his life - from 1709 to 1716 - were embittered by the long controversy with John Keill, Newton, and others, as to whether he had discovered the differential calculus independently of Newton's previous investigations, or whether he had derived the fundamental idea from Newton, and merely invented another notation for it. The controversy occupies a place in the scientific history of the early years of the eighteenth century.

The ideas of the infinitesimal calculus can be expressed either in the notation of fluxions or in that of differentials. The former was used by Newton in 1666, but no distinct account of it was printed till 1693. The earliest use of the latter in the note-books of Leibniz may probably be referred to 1675, it was employed in the letter sent to Newton in 1677, and an account of it was printed in the memoir of 1684 described below. There is no question that the differential notation is due to Leibniz, and the sole question is as to whether the general idea of the calculus was taken from Newton or discovered independently.

The case in favour of the independent invention by Leibniz rests on the ground that he published a description of his method some years before Newton printed anything on fluxions, that he always alluded to the discovery as being his own invention, and that for some years this statement was unchallenged; while of course there must be a strong presumption that he acted in good faith. To rebut this case it is necessary to show (i) that he saw some of Newton's papers on the subject in or before 1675, or at least 1677, and (ii) that he thence derived the fundamental ideas of the calculus. The fact that his claim was unchallenged for some years is, in the particular circumstances of the case, immaterial.

That Leibniz saw some of Newton's manuscripts was always intrinsically probable; but when, in 1849, C. J. Gerhardt examined Leibniz's papers he found among them a manuscript copy, the existence of which had been previously unsuspected, in Leibniz's handwriting, of extracts from Newton's De Analysi per Equationes Numero Terminorum Infinitas (which was printed in the De Quadratura Curvarum in 1704), together with the notes on their expression in the differential notation. The question of the date at which these extracts were made is therefore all important. It is known that a copy of Newton's manuscript had been sent to Tschirnhausen in May, 1675, and as in that year he and Leibniz were engaged together on a piece of work, it is not impossible that these extracts were made then. It is also possible that they may have been made in 1676, for Leibniz discussed the question of analysis by infinite series with Collins and Oldenburg in that year, and it is a priori probable that they would have then shown him the manuscript of Newton on that subject, a copy of which was possessed by one or both of them. On the other hand it may be supposed that Leibniz made the extracts from the printed copy in or after 1704. Leibniz shortly before his death admitted in a letter to Abbot Antonio Conti that in 1676 Collins had shown him some Newtonian papers, but implied that they were of little or no value, - presumably he referred to Newton's letters of June 13 and Oct. 24, 1676, and to the letter of Dec. 10, 1672, on the method of tangents, extracts from which accompanied the letter of June 13, - but it is remarkable that, on the receipt of these letters, Leibniz should have made no further inquiries, unless he was already aware from other sources of the method followed by Newton.

Whether Leibniz made no use of the manuscript from which he had copied extracts, or whether he had previously invented the calculus, are questions on which at this distance of time no direct evidence is available. It is, however, worth noting that the unpublished Portsmouth Papers (http://www.newtonproject.ic.ac.uk/portsmouth.html) show that when, in 1711, Newton went carefully into the whole dispute, he picked out this manuscript as the one which had probably somehow fallen into the hands of Leibniz. At that time there was no direct evidence that Leibniz had seen this manuscript before it was printed in 1704, and accordingly Newton's conjecture was not published; but Gerhardt's discovery of the copy made by Leibniz tends to confirm the accuracy of Newton's judgement in the matter. It is said by those who question Leibniz's good faith that to a man of his ability the manuscript, especially if supplemented by the letter of Dec. 10, 1672, would supply sufficient hints to give him a clue as to the methods of the calculus, though as the fluxional notation is not employed in it anyone who used it would have to invent a notation; but this is denied by others.

There was at first no reason to suspect the good faith of Leibniz; and it was not until the appearance in 1704 of an anonymous review of Newton's tract on quadrature, in which it was implied that Newton had borrowed the idea of the fluxional calculus from Leibniz, that any responsible mathematician questioned the statement that Leibniz had invented the calculus independently of Newton. (In 1699 Duillier had accused Leibniz of plagiarism from Newton, but Dullier was not a person of much importance) It is universally admitted that there was no justification or authority for the statements made in this review, which was rightly attributed to Leibniz. But the subsequent discussion led to a critical examination of the whole question, and doubt was expressed as to whether Leibniz had not derived the fundamental idea from Newton. The case against Leibniz as it appeared to Newton's friends was summed up in the Commercium Epistolicum (http://www.maths.tcd.ie/pub/HistMath/People/Newton/CommerciumAccount/) issued in 1712, and detailed references are given for all the facts mentioned.

No such summary (with facts, dates, and references) of the case for Leibniz was issued by his friends; but Johann Bernoulli attempted to indirectly weaken the evidence by attacking the personal character of Newton; this was in a letter dated June 7, 1713. The charges were false, and when pressed for an explanation of them, Bernoulli most solemnly denied having written the letter. In accepting the denial Newton added in a private letter to him the following remarks, which are interesting as giving Newton's account of why he was at last induced to take any part in the controversy. "I have never," said he, "grasped at fame among foreign nations, but I am very desirous to preserve my character for honesty, which the author of that epistle, as if by the authority of a great judge, had endeavoured to wrest from me. Now that I am old, I have little pleasure in mathematical studies, and I have never tried to propagate my opinions over the world, but I have rather taken care not to involve myself in disputes on account of them."

Leibniz's defence or explanation of his silence is given in the following letter, dated April 9, 1716, from him to Conti. "Pour rpondre de point en point l'ouvrage publi contre moi, il falloit entrer dans un grand dtail de quantit de minutis passes il y a trente quarante ans, dont je ne me souvenois gure: il me falloit chercher mes vieilles lettres, dont plusiers se sont perdus, outre que le plus souvent je n'ai point gard les minutes des miennes: et les autres sont ensevelies dans un grand tas de papiers, que je ne pouvois dbrouiller qu'avec du temps et de la patience; mais je n'en avois gure le loisir, tant charg prsentement d'occupations d'une toute autre nature."

The death of Leibniz in 1716 only put a temporary stop to the controversy which was bitterly debated for many years later. The question is one of difficulty; the evidence is conflicting and circumstantial; and every one must judge for himself which opinion seems most reasonable. Essentially it is a case of Leibniz's word against a number of suspicious details pointing against him. His unacknowledged possession of a copy of part of one of Newton's manuscripts may be explicable; but the fact that on more than one occasion he deliberately altered or added to important documents (ex. gr. the letter of June 7, 1713, in the Charta Volans, and that of April 8, 1716, in the Acta Eruditorum), before publishing them, and, what is worse, that a material date in one of his manuscripts has been falsified (1675 being altered to 1673), makes his own testimony on the subject of little value. It must be recollected that what he is alleged to have received was rather a number of suggestions than an account of the calculus; and it is possible that as he did not publish his results of 1677 until 1684, and that as the notation and subsequent development of it were all of his own invention, he may have been led, thirty years later, to minimize any assistance which he had obtained originally, and finally to consider that it was immaterial. During the eighteenth century the prevalent opinion was against Leibniz, but today the majority of writers incline to think it more likely that the inventions were independent.

Calculator

Leibniz constructed the first mechanical calculator capable of multiplication and division. He also developed the modern form of the binary numeral system, used in digital computers. Some find it interesting to consider what might have resulted if Leibniz had combined his findings in binary arithmetic with those developments he made in mechanical calculation.

The calculus

Although there is some question of original authorship, Leibniz is credited along with Isaac Newton with inventing the infinitesimal calculus in the 1670s. According to his notes, a critical breakthrough in his work here occurred on November 11, 1675, when he demonstrated integral calculus for the first time to find the area under the y=x function. He introduced several notations used in calculus to this day, for instance the integral sign ∫ representing an elongated S from the Latin word summa and the d used for differentials from the Latin word differentia.

Symbolic thought

Leibniz thought symbols to be very important for the understanding of things. He also tried to develop an alphabet of human thought, in which he tried to represent all fundamental concepts using symbols and combined these symbols to represent more complex thoughts, an undertaking which he never completed. A related concept is mathesis universalis. Toki Pona is an example of a modern constructed language with the same idea.

Metaphysics

His philosophical contribution to metaphysics is based on the Monadology, which introduces Monads as "substantial forms of being", which are akin to spiritual atoms, eternal, indecomposable, individual, following their own laws, not interacting ("windowless") but each reflecting the whole universe in pre-established harmony (a historically noteworthy expression of panpsychism). In the way sketched above the notion of a monad solves the problem of the interaction of mind and matter that arises in Ren Descartes' system, as well as the individuation that seems problematic in Baruch Spinoza's system, which represents individual creatures as mere accidental modifications of the one and only substance.

Theodicy and optimism

The Thodice tries to justify the apparent imperfections of the world by claiming that it is optimal among all possible worlds. It must be the best possible and most balanced world, because it was created by a perfect God.

The statement that "we live in the best of all possible worlds" was regarded as amusing by Leibniz' contemporaries, notably Voltaire who found it so absurd that he parodized him in his novel Candide, where Leibniz appears as a certain Dr. Pangloss. This parody is the root of the term "panglossianism", which refer to people holding the view that we live in the best of all worlds.

Leibniz is believed to be the first person to suggest that the concept of feedback was useful for explaining many phenomena in many different fields of study.

Leibniz's work on formal logic

The principles of the logic of Leibniz, and consequently of his whole philosophy, reduce to two:

  1. All our ideas are compounded of a very small number of simple ideas which form the alphabet of human thought.
  2. Complex ideas proceed from these simple ideas by a uniform and symmetrical combination which is analogous to arithmetical multiplication.

With regard to the first principle, the number of simple ideas is much greater than Leibniz thought; and, with regard to the second principle, logic considers three operations -- which are now known as logical multiplication, logical addition, and negation -- instead of only one.

Characters were, with Leibniz any written signs, and "real" characters were those which represent ideas directly—as the Chinese ideography was thought to—and not the words for them. Among real characters, some simply serve to represent ideas, and some serve for reasoning. Egyptian and Chinese hieroglyphics and the symbols of astronomers and chemists belong to the first category, but Leibniz declared them to be imperfect, and desired the second category of characters for what he called his universal characteristic. It was not in the form of an algebra that Leibniz first conceived his characteristic, probably because he was then a novice in mathematics, but in the form of a universal language or script. It was in 1676 that he first dreamed of a kind of algebra of thought, and it was the algebraic notation which then served as model for the characteristic.

Leibniz attached so much importance to the invention of proper symbols that he attributed to this alone the whole of his discoveries in mathematics. And, in fact, his infinitesimal calculus affords a most brilliant example of the importance of, and Leibniz's skill in devising, a suitable notation.

Universal characteristic and calculus ratiocinator

What is usually understood by the name "symbolic logic", is what Leibniz called a calculus ratiocinator and is only a part of the Universal Characteristic. In symbolic logic Leibniz enunciated the principal properties of what we now call conjunction, disjunction, negation, identity, set-inclusion, and the empty set; but the aim of Leibniz's researches was, as he said, to create "a kind of general system of notation in which all the truths of reason should be reduced to a calculus. This could be, at the same time, a kind of universal written language, very different from all those which have been projected hitherto; for the characters and even the words would direct the reason, and the errors -- excepting those of fact -- would only be errors of calculation. It would be very difficult to invent this language or characteristic, but very easy to learn it without any dictionaries". He fixed the time necessary to form it: "I think that some chosen men could finish the matter within five years"; and finally remarked: "And so I repeat, what I have often said, that a man who is neither a prophet nor a prince can never undertake any thing more conducive to the good of the human race and the glory of God".

In his last letters he remarked: "If I had been less busy, or if I were younger or helped by well-intentioned young people, I would have hoped to have evolved a characteristic of this kind"; and: "I have spoken of my general characteristic to the Marquis de l'Hpital and others; but they paid no more attention than if I had been telling them a dream. It would be necessary to support it by some obvious use; but, for this purpose, it would be necessary to construct a part at least of my characteristic; -- and this is not easy, above all to one situated as I am".

Leibniz thus formed projects of both what he called a characteristica universalis, and a calculus ratiocinator. It is not hard to see that these projects are interconnected, since a perfect universal characteristic would comprise a logical calculus. Leibniz did not publish the incomplete results which he had obtained, and consequently his ideas had no continuators, with the exception of Lambert and some others, up to the time when Boole, De Morgan, Schrder, MacColl, and others rediscovered his theorems. Frege remarked that his own symbolism is meant to be a calculus ratiocinator as well as a lingua characteristica.

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