Lecture Supplement Introducing Leibniz

     Copyright © 2014 Bruce W. Hauptli




1. Leibniz’ Life and Interests [1646-1716]:


Gottfried Wilhelm Leibniz’ father was a Professor of Moral Philosophy at the University of Leipzig, and as a young child he was largely self-taught from the works in his father’s library.  His course of studies centered upon the Aristotelian and scholastic “classics.”  He entered the University of Leipzig at the age of 15 as a law student, and there learned something about the revolutionary ideas of Galileo, Bacon, Hobbes, and Descartes.  He concluded his studies in five years, but was not granted his degree because of his age.[1]  He then attended the lesser-known University of Altdorf, and earned a degree in law and philosophy in 1666.  While he was offered a faculty position there, he entered a life of public service (like Hobbes and Locke in England), and occupied a number of positions for noble men throughout his life. 


     Diplomatic assignments took Leibniz to Paris in 1672 (he also visited England, and met with Spinoza on his way home in 1676), and on these travels he became even more familiar with the “modern thinkers.”  Leibniz entered into a wide correspondence with many of the major thinkers of his day, and he began publishing in important intellectual forums such as the Journal des Savants. 


     Whereas Descartes and Spinoza were motivated to “build” intellectual systems of thought which would clearly and distinctly display certain “core ideas,” Leibniz is best seen as a broad-ranging thinker whose primary wish was to bring many disparate elements together: Christians of all sects, medical researchers, scientific researchers, legal theorists, and mathematicians.  It is the latter field which is central to Leibniz’ thought, however (and this should not be surprising when we remember that he is one of two individuals credited with the development of calculus).  Like the other “continental rationalists,” Leibniz maintained that reality is not only rational at its core, but like them he held that mathematics helped us understand this rational core. 


     Perhaps we should pause and consider this point so that we don’t misattribute our views to those of the Early Modern thinkers.  When we think of science, we do not generally consider mathematics a science—after all, it is an a priori rather than an empirical (a posteriori), discipline.  But thinkers like Descartes, Spinoza, and Leibniz thought of mathematics as not only a science, but as the science.  In short they thought held that “what is real is rational” while holding also that “mathematics is the paradigm of rationality,” and this leads fairly directly (as we can see most clearly in Spinoza) to “reality is mathematical.”  In his Leibniz’s New Essays Concerning Human Understanding: A Critical Exposition, John Dewey notes that:


in the applicability of mathematics to the interpretation of nature Leibniz finds witness to the continuity and order of the world.  We have become so accustomed to the fact that mathematics may be directly employed for the discussion and formulation of physical investigations that we forget what is implied in it.  It involves the huge assumption that the world answers to reason; so that whatever the mind finds to be ideally true may be taken for granted to be physically true also.  But in those days, when the correlation of the laws of the world and the laws of mathematical reasoning was a fresh discovery, this aspect of the case would not be easily lost sight of.[2] 


Where geometry was the paradigm mathematical field for the other rationalists, however, Leibniz is perhaps not best thought of along geometric lines.  He did not begin with certain foundational beliefs in the manner of Descartes or Spinoza.  Where Spinoza was concerned with working out the deductive consequences of the basic definition of substance, and Descartes was concerned with discovering the foundations of knowledge, Leibniz wanted to unify a large number of different areas of thought.  He wanted to resolve paradoxes, problems, and inconsistencies by developing a coherent, unified, overall theory that all individuals could rationally accept.  As Dewey notes,


Leibniz was a man of his times...and was himself actively engaged in the prosecution of mathematics, mechanics, geology, comparative philology, and jurisprudence.  But he was also a man of Aristotle’s times,—that is to say, a philosopher, not satisfied until the facts, principles and methods of science had received an interpretation which should explain and unify them.[3] 


It is thus ironic, as Dewey notes, that Leibniz was involved in so many divisive intellectual conflicts:


it is somewhat significant that one whose tendency was conciliatory, who was eminently what the Germans delight to call him, a “mediator,” attempting to unite the varied truths which he found scattered in opposed systems, should have had so much of his work called forth by controversy.  Aside from the cases just mentioned [contra the Newtonians regarding the infinitesimal calculus; and contra Bishop Clarke regarding the nature of God, time, space, and freedom], his other chief work, the Theodicy, is, in form, a reply to Bayle....But Leibniz has a somewhat different attitude towards his British and towards his Continental opponents.  With the latter he was always in sympathy, while they in turn gave whatever he uttered a respectful hearing.  Their mutual critiques begin and end in compliments.  But the Englishmen found the thought of Leibniz “paradoxical” and forced.  It seemed to them wildly speculative, and indeed arbitrary guess-work, without any special reason for its production, and wholly unverifiable in its results....But Leibniz, on the other hand, felt as if he were dealing, in philosophical matters at least, with foemen hardly worthy of his steel.  Locke, he says, had subtlety...and a sort of superficial metaphysics; but he was ignorant of the method of mathematics,—that is to say, from the standpoint of Leibniz, of the method of all science.[4] 


Nonetheless, Leibniz was a firm believer in harmony, and this meant that he looked for every opportunity to try to bring conflicting orientations together.  While he is interested in so many different philosophical problems, if we are to understand his orientation, we must pay special attention to two particular problems (in part because they are not familiar to us): (i) the problem of the continuum and motion and (ii) the problem of fate and freedom. 


     Our understanding of Leibniz is hampered by the fact that there is no central canonical text in his corpus.  As Roger Ariew and Daniel Garber note in their “Principle of Selection and Rational for the Volume,” to their G.W. Leibniz: Philosophical Essays:


there is nothing in Leibniz’s enormous corpus that corresponds to Descartes’s Meditations, Spinoza’s Ethics, or Locke’s Essay, no single work that stands as a canonical expression if its author’s whole philosophy.  Although works like the “Discourse on Metaphysics” and the “Monadology” are obviously essential to any good collection of Leibniz’ writings, neither of these nor any other single work is, by itself, an adequate exposition of Leibniz’s complex thought.  Unlike his more systematic contemporaries, Leibniz seems to have chosen as his form the occasional essay, the essay or letter written about a specific problem, usually against a specific antagonist, and often with a specific audience in mind.[5] 


2. The Problem Of The Continuum and Motion:


The Pythagoreans discovered that the side of the unit square is incommensurable with its diagonal—they discovered what we call irrational numbers.[6]  This led to problems because these numbers could not be assigned a place on the number line and could not be readily fit into the Greek view of mathematics which emphasized the study of proportions (e.g., ratios of rational numbers).  Without a clear understanding of infinity, however, paradoxes quickly develop as one talks about the relationship of the rationals and irrationals.  Aristotle did not countenance the notion of an actual or completed infinite totality, however, and he maintained that a totality could only be potentially infinite—its finite membership may be increased without limit but this does not mean there actually is a completed infinitude. 


     Leibniz, on the other hand relied fundamentally upon the idea of a completed infinitude.  He held to a principle of continuity maintaining that everything in nature happens by degrees and that there are no discontinuities or leaps in natural occurrences.  This was important to him as he grappled with the concept of motion.  Throughout our Western history this has been an important concept.  In 600-500 B.C.E., early Greek philosophers like Anaximander and Heraclitus took motion to be the essential characteristic of the natural world.  They did not inquire into its nature however.  In 500-400 B.C.E., Eleatic followers of Parmenides maintained that a close philosophical analysis showed that motion is merely an apparent phenomenon.  Followers like Zeno soon used various paradoxes to argue that motion was illusory: “traversing an infinitude of points takes an infinitude of time,” “a body can move neither where it is nor where it is not and, therefore, motion is impossible—a merely apparent phenomenon,” etc. 


     Aristotelian science treated motion as a change from a potentiality to an actuality.  That is, it was treated qualitatively and this makes any mathematical study of motion most difficult—the most that can be done here is to speak of ratios and proportions.  For Aristotle “all things that move are moved by something else.”  He maintains that motion is the actualization of what exists potentially—a change in the qualities of things:


Aristotle’s kinematics, like his physics in general, was a qualitative science, incapable of providing a precise definition of such motions as velocity and acceleration.  In fact, Greek mathematics, with its insistence on the illegitimacy of proportions or ratios between heterogeneous qualities, did not provide even the formal means of defining velocity as the ratio between distance and time....[7] 


     Over time a problem arose in regard to the increase or decrease of qualities—the question was how qualities like warmness or blackness could vary in their intensities.  As something changes its qualities, it must change from one quality to another but this seems to imply an infinitude of qualities between the grosser ones we usually note (hot and cold, white and black).  Another problem was whether motion involved two qualities (the change of one and the acquisition of another—the terminus or end) or whether it involved only a gradual change of one quality.  A notable implication of Aristotle’s view is that it seemed to require that the earth not move if the heavens were to move—there must be something that is unmoved. 


     By Galileo’s time we are back to the early Greek Atomists’ view of bodies in motion.  An important distinction to Copernicus is that between relative and absolute rest and motion.  This allowed physicists to deal with the apparent rest of the earth while accepting the traditional idea that movement required some fixed frame of reference.  Descartes recognizes the importance of this distinction and would distinguish between vulgar views of motion (change of place) and a scientific view (transfer of matter from the vicinity of those bodies which it is in immediate contact into the vicinity of other bodies). 


     As Roger Penrose notes,


the profound breakthrough that the seventeenth century brought to science was the understanding of motion.  The ancient Greeks had a marvelous understanding of things static—rigid geometrical shapes, or bodies in equilibrium (i.e. with all forces balanced, so there is no motion)—but they had no good conception of the laws governing the way that bodies actually move.  What they lacked was a good theory of dynamics, i.e. a theory of the beautiful way in which Nature actually controls the change in location of bodies from one moment to the next.  Part (but by no means all) of the reason for this was an absence of any sufficiently accurate means of keeping time, i.e. of a reasonably good ‘clock’.  Such a clock is needed so that changes in position can be accurately timed, and so that the speeds and accelerations of bodies can be well ascertained.  Thus Galileo’s observation in 1583 that a pendulum could be used as a reliable means of keeping time had a far-reaching importance for him (and for the development of science as a whole!) since the timing of motion could then be made precise.  Some fifty-five years later, with the publication of Galileo’s Discoursi in 1638, the new subject of dynamics was launched—and the transformation from ancient mysticism to modern science had begun![8] 


3. Contrasts Between Descartes, Spinoza, and Leibniz:


Descartes maintained that:


all the variety in matter, or all the diversity of its forms depends on motion.  There is therefore but one matter in the whole universe, and we know this by the simple fact of its being extended.  All the properties which we clearly perceive in it may be reduced to the one, viz. that it can be divided, or moved according to its parts, and consequently is capable of all these affections which we perceive can arise from the motion of its parts.  For its partition by thought alone makes no difference to it; but all the variation in matter, or diversity in its forms, depends on motion.[9] 


This view is one that becomes popular as the study of dynamics provides the “direction” for the development of the physical sciences.  Whatever the merits of Descartes’ physical theories, however, his philosophical world-view bifurcates the universe and runs into at least the following problems:


mind/body interaction,

different causal chains (for the mental and the physical),


the number and nature of substance(s), and



     As we have seen, Spinoza sought to avoid these difficulties with his talk of the cause sui.  As we also saw, however, significant problems arise with his system (including at least the following):  


lack of freedom (nothing in the universe is contingent),

monism (only one substance), and

rejection of teleology and final causes. 


Leibniz’s philosophical system is significantly different from both of these.  He holds that there is both a transcendent deity and a created universe, and he holds that there are an infinite number of substances (or monads).  He endeavors to avoid both the problems which Descartes has regarding the interaction of substances in the created world, and the problems which Spinoza has with determinism.  He tries to build a harmonious world-system which gives us a fully deductive, completely rational, created world which is fully explained by a necessarily existing deity who freely creates a world which accords with the mechanistic physical laws while allowing for free action and activity. 


4. Leibniz’ Conception of Substance:


Thus Leibniz develops a “third” orientation:


like Spinoza and Descartes (and unlike Pascal), he views the universe as a harmonious, rational whole.  Unlike Spinoza, he asserts there are many substances (indeed, an infinite number of them) and he speaks of a universal harmony of this infinitude.  He felt that both Descartes and Spinoza were wrong to assume that extended things were real (whether one treats them as substances or as attributes).  Once one makes extension real, he claims, one is forced to divide the world into two substances or attributes and this makes it difficult to relate these aspects of the universe to one another.  Leibniz hoped that he could show that extension, figure, and motion (the things appealed to explain events in the “physical world”), as well as thoughts could be explained as derivative—that he could find some concept that could be used to explain both the physical events and the mental ones. 


His studies confirmed him in his opinion that a different sort of account of substance was needed—bare extension, the essential characteristic of the physical according to the others (whether treated substantially or as an attribute), could not explain motion by itself.  Galilean mechanics explained motion in terms of successive states where a body occupies a different position in each moment.  As the science become more and more satisfactory, it became necessary to appeal to concepts like continuously increasing velocity, infinitesimal times and distances, etc.  Leibniz and Newton devised calculus to deal with the attendant problems that such talk engendered (if one treats motion as analyzable into successive discrete states, one must allow natural objects to make “leaps” from one spot to another).  As long as one treats nature as discrete, Leibniz believed, one can not escape the problems of the “continuum:”


now this is the axiom which I utilize, namely, that “no event takes place by a leap.”  This proposition flows...from the laws of order and rests on the same rational ground by virtue of which it is generally recognized that motion does not occur by leaps, that is, that a body in order to go from one place to another must pass through definite intermediate places.... 

  I do not believe extension alone constitutes substance, since its conception is incomplete....For we can analyze it into a plurality, continuity and co-existence....I believe that our thought of substance is perfectly satisfied in the conception of force and not in that of extension....[10] 


In his A New System of Nature, Leibniz says that:


but since having tried to lay the foundations of the very principles of mechanics in order to give a rational account of the laws of nature known to us by experiment, I realized that the sole consideration of an extended mass did not suffice, and that we must again employ the notion of force which is very intelligible despite its springing from metaphysics.....it is impossible to find the principles of a true unity in matter alone or in that which is only passive, since everything in it is only a collection or mass of parts to infinity....the continuum cannot be composed of points.  Therefore...I was compelled to have recourse to a formal atom since a material being cannot be both material and perfectly indivisible or endowed with a true unity.  It was necessary, hence to recall...the substantial forms so decried today....their nature consists in force....[11] 


That is, the difficulties associated with the notion of a continuum were instrumental in leading Leibniz to the view that motion itself can not be a real entity because it never exists and only what exists in time can be real.  Where many felt physical reality could be fully explained by appeal to extension, figure, and motion, Leibniz came to believe these were apparent rather than real, and that the basic universal laws of nature could not be derived from such concepts.  In his “Introduction” to a collection of Leibniz’ essays, Paul Janet provides a discussion which helps us come to understand Leibniz’ objection to the notion of an extended substance:


...I think it can be shown a priori that matter taken in itself is something ideal and super-sensible, at least to those who admit a divine intelligence.  For it will readily be granted that God does not know matter by means of the senses; for it is an axiom in metaphysics that God has no senses and consequently cannot have sensations.  Thus: God can be neither warm nor cold; he cannot smell the odor of flowers; he cannot hear sounds, he cannot see colors; he cannot feel electrical disturbances, etc.  In a word, since his is a pure intelligence he can conceive only the purely intelligible; not that he is ignorant of any of the phenomena of nature, only that he knows them in their intelligible reasons and not through their sensible impressions, by means of which creatures are aware of them.  Sensibility supposes a subject with senses, organs and nerves, that is, it is a relation between created things.  From God’s point of view, therefore, matter is not sensible; it is, as the Germans say, ubersinnlich.  The conclusion is easy to draw, namely, that God, being absolute intelligence, necessarily sees things as they are, and conversely the things in themselves are such as he sees them.  Matter is, accordingly, such in itself as God sees it, but he sees it only in its ideal and intelligible essence, whence we see that matter is an intelligible something and not something sensible.[12] 


5. Leibniz’ Monads:


Leibniz calls his basic substances “monads” (from the Greek ‘monas’ meaning “unity” or “that which is one”)—“metaphysical units” of force (“centers of psychic activity”).  Like Spinoza (and unlike Descartes) he holds there is no possible interaction of substances and, thus, for him the harmony between substances must be imposed from without.  According to Leibniz, substances can not interact with one another, but in a rationally-ordered world there must be correlations between the various predicates which are true of one substance and those which are true of another.  If, for example, it is to be true of one substance, or monad, (call it Bruce) that it is married, then there must be another monad (call it Laurie) of which it is true that it is married to Bruce.  The truth of the first proposition requires that there be another true proposition of the second sort. 


     For Leibniz, if the basic substances interacted, they would not truly be substances (here we can see that he agrees with Spinoza, who argues that if one substance caused something to be true of a second one, that second one would not be a substance—it would be “conceived through another”).  Like Spinoza, then, he believed that a major defect of Cartesian bare extended matter was that it was conceived of as inactive—it could not explain why anything happened and the deity had to be brought into the picture as a constant source of action (Descartes’ deity conserves as well as creates substances).  Leibniz felt we should consider activity to be a basic characteristic of substances: “substance is a being capable of action.”[13]  While Leibniz sees Spinoza as preferable to Descartes on these points, he rejects Spinoza’s monism! 


     Both Descartes and Spinoza thought of substance as a subject in the grammatical or logical sense—the attributes become the true predications of substance, and it is supposed that these predicates can not exist without the subject (though the subject could exist without the predicates).  Besides the logical/grammatical notion of a subject, however, there is also the notion of an element that persists through change.  While Leibniz agrees with Descartes and Spinoza that substances can only be subjects and never be predicated of another thing, he holds that it is just as important that they are “capable of action.” 


     Neither Descartes nor Spinoza developed a clear conception of propositions and truth however.  For this reason, Leibniz’ “theory of propositions” and his “theory of truth” are important.  He held that the subject-predicate form of proposition was basic—that all true propositions are subject-predicate ones.  Moreover, to ensure that substances have the sort of “integrity” which he felt was basic to them, he held that they could never be properly placed in the predicate proposition.  For him, relational propositions (like “Bruce and Laurie are married”) can not be real propositions then, they must be analyzed into subject-predicate sentences (“Bruce is married” and “Laurie is married”).  Leibniz holds that a substance is a logical subject of which many things may be attributed, but which can not itself be attributed (or predicated) of anything else.  Moreover, he held that the concept of the subject contains all the predicates that will be true of it. 


     His analysis of truth held that what makes a sentence true is that the concept of the predicate of the proposition is contained within the concept of the subject.  Thus, really, all sentences are analytic.[14]  Leibniz held, however, that some are finitely analytic (e.g., “A male is a male,” or “My brother is a male”), and that some are infinitely analytic (e.g., “Julius Caesar died in 44 B.C.E.). 


     According to Leibniz, monads have an internal principle of change—he calls it “appetition or desire.”  Monads are the fundamental real things, the subjects of change which contain within themselves the principle of their changes.  His view of the universe is that it is filled with monads each reflecting the universe from a different perspective. 


6. The Role of Leibniz’ Deity:


Leibniz’ deity plays an important role in his system for two reasons.  First the deity causes the individual substances (monads), and provides for the pre-established harmony between the changes of these substances; and, second, it is through the deity that we get freedom or teleology in the world. 


     To have independent substances which act solely from their own nature, Leibniz believes that he must allow that “...the notion of an individual substance includes once and for all everything that can ever happen to it....”[15]  But if this is the case, how can there be freedom and teleology?  Well, he would avoid the deterministic picture of the universe which Spinoza offers while holding on to the truth Spinoza accepts that substances can not interact causally with one another without ceasing to be substances.  He does this by allowing that the events in the one causal chain which is the harmonizing thread which runs through the universe are “certain, but not necessary.” 


7. Leibniz’ Central Principles:


To understand Leibniz, we will need to distinguish the various central principles that are at the core of his theories:


the principle of contradiction (or of non-contradiction),


the principle of sufficient reason,


the principle of perfection (or of the best). 


Central to these principles is his distinction between truths of reason and truths of fact—but while his distinction is somewhat like the one that is drawn now, we should be careful not to attribute our view to Leibniz!  For him, truths of reason are self-evident truths of reason, or they are easily reducible to such truths.  Except for the case of the deity’s existence, no existential propositions are on this list.  The denials of such truths are contradictions and, thus, can not but be true.  Thus, these truths are guaranteed by the principle of contradiction.  Truths of fact, on the other hand, require some sufficient reasonthere must be some explanation as to why they are the case (are true).  Here we might distinguish the possible from the actual—some truths of fact are not true because others are (they are possible but nonactual)—it will be convenient to speak of essences here.  The things of the world (the monads) have certain essences or natures (certain things are true of them).  They need not have been the way they are, since other essences are noncontradictory.  Why, then, is the world the way it is?  For Leibniz, the sufficient reason for the world being the way it is is the principle of the Best! 


Truths of Reason

Truths of Fact (possible)

Truths of Fact (actual)

Principle of Non-Contradiction

Principle of Sufficient Reason

Principle of the Best

Statements about essences

Possible Worlds

Co-possibles (Sphere of the Possible)

The Actual World

God could have chosen a different world (he is metaphysically free yet morally constrained

Finitely Analytic


Infinitely Analytic



     In addition, there are two other centrally important principles for Leibniz:


the principle of the identity of indiscernibles: what distinguishes individual substances?  Only their characteristics/attributes. 


the principle of continuity: there are no holes in the universe. 


     To further understand Leibniz’ world-view, we need to discuss the phenomenal character of both causation and composites, but this introduction is already over-long, and, instead, we will turn to the texts. 



Notes: (click on note number to return to the text for the note)

[1] Brief biographies hint at some no longer understood intrigue, but don’t identify it, while other sources give the above explanation, which I accept as the most reasonable. 

[2] John Dewey, Leibniz’ New Essays Concerning Human Understanding: A Critical Exposition [1888], the citations are from selections reprinted in From Plato to Wittgenstein: The Historical Foundations of the Modern Mind, ed. Daniel Kolak (Belmont: Wadsworth, 1994), pp. 317-360, pp. 323-324. 

[3] Ibid., p. 319. 

[4] Ibid., p. 334. 

[5] Roger Ariew and Daniel Garber, “Principle of Selection and Rational for the Volume,” in G.W. Leibniz: Philosophical Essays, trans. and eds. Roger Ariew and Daniel Garber (Indianapolis: Hackett, 1989), pp. x-xii, p. x. 

[6] That is, a square whose sides are “one unit” long will not have a diagonal which can be expressed as a ratio of two “natural numbers” (p/q)—as we would put it today, the diagonal has a length of the “square root of 2.” 

[7] Max Jammer, “Motion,” in The Encyclopedia of Philosophy v. 5, ed. Paul Edwards (N.Y.: Macmillan, 1967), pp. 396-399, p. 397. 

[8] Roger Penrose, The Emperor’s New Mind (New York: Oxford U.P., 1989), p. 162. 

[9] Rene Descartes, Principles of Philosophy II, 23; in The Philosophical Works of Descartes, trans. E.S. Haldane and G.R.T. Ross (Cambridge: Cambridge U.P., 1969) v. 1, p. 265.  Emphasis added to passage. 

[10] Letter to De Volder in Leibniz: Selections, ed. P.P. Wiener (New York: Scribners’, 1951), pp. 157-158.  Emphasis added to passage twice. 

[11] G.W. Leibniz, New System of Nature [1695], in The Philosophy of the 16th and 17th Centuries, ed. R.H. Popkin (N.Y.: Macmillan, 1966), pp. 324-325.  Also in G.W. Leibniz: Philosophical Essays, eds. Roger Ariew and Daniel Garber, op. cit., pp. 138-145, p. 139.  Emphasis added to passage four times. 

[12] Paul Janet, “Introduction,” in Leibniz: Discourse on Metaphysics, Correspondence with Arnauld, and Monadology [1902], trans. George Montgomery (La Salle: Open Court, 1968), pp. vii-xxiii, pp. xvii-xviii. 

[13] Leibniz G.W., Principles of Nature and of Grace, Based on Reason [1714], in The Light of Reason, ed. Martin Hollis (London: Fontana/Collins, 1973), p. 335. 

[14] An analytic statement is one that is supposed to be true in terms of the mere meaning of the terms involved.  For example “squares are four-sided figures” is true because (a) ‘square’ means ‘four [equal]-sided figure’, and, thus, it reduces to an “identity statement.”  Such truths are conceptual truths.  These sorts of statements are to be contrasted with synthetic ones—they are truths that are not true simply in terms of the meanings of the terms involved.  Here the truth of the statement depends on the way the world is, rather than, simply, on the way words are used.  The distinction between analytic and synthetic truths or statements is, primarily, a semantic one. 

[15] Leibniz, Discourse on Metaphysics, XIII, in Leibniz, trans. G.P. Montgomery (Chicago: Open Court, 1968), p. 19. 

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Last revised on: 11/04/2014