Gottfried Wilhelm Leibniz
Mr. Leibniz
Education: Leipzig University (1661–1666:

BA in phil., Dec. 1662; MA in phil., Feb. 1664; LL.B., Sep. 1665; Dr.phil.habil., Mar. 1666) University of Jena (summer school, 1663)[1] University of Altdorf (Dr.jur., Nov. 1666)

Noteable Ideas: Calculus

Monads Best of all possible worlds Leibniz formula for π Leibniz harmonic triangle Leibniz formula for determinants Leibniz integral rule Principle of sufficient reason Diagrammatic reasoning Notation for differentiation Proof of Fermat's little theorem Kinetic energy Entscheidungsproblem AST Law of Continuity Transcendental Law of Homogeneity Characteristica universalis Ars combinatoria Calculus ratiocinator Universalwissenschaft

Main Interests Mathematics, metaphysics, logic, theodicy, universal language
Born July 1, 1646
Died November 14, 1716 (aged 70)

Gottfried Wilhelm Leibniz  (July 1, 1646 – November 14, 1716) was a German philosopher and mathematician. In regards to the former role, Leibiniz was one of the three greatest advocates of rationalism in the 17th Century, along with Rene Descartes and Baruch Spinozoa. His philosophy is mostly noted for its theistic optimism: i.e., for his conclusion that the universe we find ourselves in was the best of all worlds that God could have created. In regards to the latter role, Leibniz developed the calculus of infinitesmals contemporaneous yet indepedently of Isaac Newton, and his clear mathematical notion is generally used in calculus textbooks even to this today (in lieu of Newton's rather more complicated, deliberately obscure form of notation).

Fundamental Principles

(1) The Principle of Contradiction (PIC): " A proposition cannot be true and false at the same time; therefore, A is A and cannot be not-A."

(2) The "Predicate-in-Notion" Principle" (PIN): "In every affirmative proposition, whether necessary or contingent, universal or particular, the notion of the predicate is in some way included in that of the subject," as Leibiniz put it. 

Leibniz strikingly combined the PC with the PIN principle (both borrowed in their original form from Aristotle) in his philosophical system, asserting in his Primary Truths that "all remaining truths are reduced to primary truths with the help of definitions, that is, through the resolution of notion". Even more strikingly, the combination of PC and PIN entails that since in any true proposition the predicate is contained explicitly or implicitly within the subject, this is so for all affirmative truths, whether they be universal or particular, necessary or contingent.

(3) The Principle of Sufficient Reason (PSR): "Nothing is without a cause, or reason why." According to Leibniz, this principle actually follows from the PIN principle, in the following manner: if there were a truth that had no reason, then there would be a proposition whose subject did not contain the predicate; however, this violates the PIN principle, therefore everything has a cause, or reason why it is the way it is. He suggests in his Principles of Nature and Grace that the claim that nothing takes place without a sufficient reason means that nothing happens in such a way that it is impossible for someone with enough information to give a reason why it is so and not otherwise; however, he admits quite frankly elsewhere that for most of the time these reasons cannot be known to us.

Often associated (but not synonomous) with this principle in Leibniz's usage is the Principle of the Best, which essentially says that God always acts for the best. In regards to the creation of the world in Leibniz's philosophy, the proposed sufficient reason for God's choice of our particular world is that this world is the best of all possible worlds, which probably explains much of the confusion between the principles amongst the lay philosophical audience. 

(4) The Principle of the Identity of Indiscernibles (PII): One of the more controversial of Leibniz's basic philosophical principles, this principle in one of his typical formulations of it states, "it is not true that two substances can resemble each other completely and differ only in number", or (∀F)(Fx ↔ Fy) → x = y. Leibniz was adamant that spatio-temporal properties are excluded from the list of properties which can properly count as difference-making properities, asserting that there can be no purely extrinsic determintions, but that all extrinsic differences must be founded upon an instrinsic difference between two substances. 

An uncontroversial, related principle used by Leibniz is the Principle of the Indiscernibility of Identicals: if two things are identical, then they share all properties, or x = y → (∀F)(Fx ↔ Fy). The combination of these two principles is known as Leibniz's Law: two things are identical if and only if they share all properties, or x = y ↔ (∀F)(Fx ↔ Fy). (The Principle of the Indiscernibility of Identicals by itself is sometimes incorrectly referred to as Leibniz's Law.)

(5) The Principle of Continuity:"Nothing takes place suddenly, and it is one of my great and best confirmed maxims that nature never makes leaps," as he put it in the preface to his New Essays. Leibniz believed his natural law implied that every change passes through some intermediate change, and that there is an actual infinity in things. He employed this principle to show that no motion could arise from a state of complete rest, and that "noticeable perceptions arise by degrees from ones which are too minute to be noticed".

Basic Metaphysics

From Leibniz's perspective, the fundamental questions of metaphysics were essentially reducible to ontological questions, predicated on one's answer to such questions as "What is?", "What are the most basic components of reality?", and "What grounds what else?"

His answer to this question throughout life was everything is composed or reducible to simple substances; everything could be grounded in simple substances. It was his belief that each substance had a complete concept individual to itself and that each was essentially an active unity capable of perception and appetition. 

In his Discourse on Metaphysics, Leibniz claims that the Aristotlean understanding of substance as being that which is the subject of predication and which cannot be predicated of something else was insufficient to fully describing the nature of substance. Appealing to the Principle of Contradiction and the Predicate-in-Notion Principle, he then made a case that the concept of the predicate is contained in the concept of the subject in every true predication. From this, he concludes that the "nature of an individual substance or of a complete being is to have a notion so complete that it is sufficient to contain and to allow us to deduce from it all the predicates of the subject to which this notion is attributed.” Putting it differently, y is a substance if and only if has what Leibniz would call a complete individual concept (CIC), a concept that contains within it all predicates of past, present, and future; the CIC serves to individuate every finite substance out from the infinity of other finite substances, being the sum of the qualitative properties inherent within its given object. The complete individual concept of a substance is thus equivalent to the essence of the substance as it might be known by a omniscient being. Leibniz concludes from this his doctrine of marks and traces; namely, that when the connection of things is carefully considered, within a given substance there are vestiges of everthing that has happened to and marks of everything that will happen to its object (and thus, even traces of everything that happens in the universe), "even though God alone could recognize them all.”

Lebiniz derives the following consequences from his conception of substance and doctrine of marks and traces:      

(1) No two substances can resemble each other completely and be distinct. (His Principle of the Identity of Indiscernibles). If the CIC is that which allows one to individuate any substance from an infinity of substances, then if the individual concepts of substance x and do not allow us to properly distinguish x from y, then their individual concepts must be considered as yet incomplete. Stated in the converse fashion, there must always be a reason found within the complete individual concept of their substances that x is discernible from y. Thus, for every complete individual concept of essence, there can be only one matching substance instantiated in the world: Lebiniz's Principle of the Identity of Indiscernibles.

(2) A substance is not divisible & (3) One substance cannot be constructed from two.  If the CIC for substance a is sufficiently rich to allow the deduction of all its predicates past, present, and future, then the division or fusion of substances must be deducible from the pre-division/pre-combination predicates. In the case of substantial division, the new substances aand awould both have all of the predicates of a, as well as other predicates unique to each new substance; in the case of substantial recombination, the new substance ab would have all of the predicates of both a and b, except that which serves to distinguish substance a from substance b (which would therefore instantiate them as seperate substances). But in the former case, the predicates unique to aand the predicates unique to awould have to be deducible from the pre-division predicates of a, or else they would not have the attribute of simply being substance a in the past; thus, their concepts could not be distinguished from the CIC of belonging to substance a, and therefore they cannot be treated as becoming seperate substances using Leibniz's defintion.  Likewise, in the latter case, the subtraction of the distinguishing predicates for substance a from substance would have to be deducible from the pre-combination predicates of both a and b, or else they would not have the attribute of simply being the substance ab in the future; thus, their concepts could not be distinguished from the CIC of substance ab, and therefore they cannot be treated as having ever been seperate substances using Leibniz's defintion. Therefore, substances logically can neither be divided nor constructed from other substances, according to Leibiniz's understanding of the word. 

(4) A substance can only begin in creation and end in annihilation. If the CIC contains all predicates of the substance past, present, and future, then (Leibniz seems to argue) this must include truths about the subject extending back into the past and forward into the future until the ends of time. 

(5) The number of substances does not naturally increase and decrease. Given the assumptions that substances only naturally arise in creation and disappear in annhihilation, and that substances can undergo neither fission nor fusion, it immediately becomes evident that the number of substances cannot naturally increase or decrease.

(6) Every substance is like a complete world and like a mirror of God/the whole universe, which each expresses in its own way. If it is the nature of a substance to have a CIC from which all its predicates past, present, and future may be deduced, and if substances exist from the creation of the world, then each substance naturally acts as a mirror of the entire happenings of the universe from its own particular self-contained perspective. 

Yet another consequences of Leibniz's conception of substance was his denial of the causal interaction of finite substances. Leibiniz's doctrine required him to not only deny the actuality of genuine physical influx (the transfer of one property within a substance to some other property), but also affirm that the reasons for any property that a substance may have are already entailed by its CIC: every state of a substance may be explained, grounded, or caused by its own complete individual notion. 

Critique of Descartes

If a finite substance is to have a CIC, what is its ontological status? In other words, what kind of thing could have such a CIC or such a nature? Leibniz's answer to this question was that while it is the nature of an individual substance to have a CIC, only a genuine unity qualifies as a real substance. Leibniz stated as much in this manner, " “To put it briefly, I hold this identical proposition, differentiated only by the emphasis, to be an axiom, namely, that what is not truly one being is not truly one being either.” The fundamental idea is that for something to have a CIC, there must be something about it that guarantees or makes possible the unity of its substance, and this is understood as its substantial form or soul: only a soul or substantial form is the kind of thing which can be properly said to have or underlie a complete individual concept, Leibniz reasons, for only a soul or substantial form is by its nature an imperishable unity.

“A substantial unity requires a thoroughly indivisible and naturally indestructible being, since its notion includes everything that will happen to it, something which can be found neither in shape nor in motion (both of which involve something imaginary, as I could demonstrate), but which can be found in a soul or substantial form, on the model of what is called me.”

By declaring that a substance is necessarily indivisible, Leibniz makes it impossible by definition for a body or matter alone to be considered a true substance in his philosophy. A Cartesian "corporeal substance" is divisible, as its essence is simply its extension, and therefore in Leibniz's philosophy a Cartesian chunk of matter is not a substance, as he argues that nothing that is divisible can rightly be considered as a substance. Leibniz considered the case of a human body deprived of its soul, and said that the cadaver would not be considered a proper substance (and thus not as real a being as the simple substances it is composed of) at all, but rather merely an aggregate of substances. Aggregates of simple substances in Leibniz's philosophy therefore have a different ontological status from simple substances.

The distinction between simple substances and aggregate of said substances therefore attains great importance in Leibniz's system of philosophy. According to Leibniz himself, “I hold that philosophy cannot be better reestablished and reduced to something precise, than by recognizing only substances or complete beings endowed with a true unity, together with the different states that succeed one another; everything else is only phenomena, abstractions, or relations.” Some Leibniz scholars have suggested that this implies that aggregates of simple substances are merely phenomenal and fail to have the reality of the underlying simple substances, and that therefore the bodies of natural philosophy (another word for the physical/biological sciences) would then seem to be in some sense mere phenomena and thus imaginations leading us astray from the underlying metaphysical reality, with only Leibniz's metaphysical philosophy acting as a path to the true reality of the underlying simple unities. However, a more accurate conception would pay due attention to Leibniz's distinction between a real unity (unum per se) and a phenomenal unity (unum per aggregationem); Leibniz liked to draw the comparison between the latter and a rainbow, saying that material bodies fail to have intrinsic unity but we represent them as single, unified object just as we represent a rainbow as one thing when in reality it is merely the result of the refraction of light through an aggregate of unnumbered water droplets. Continuing the metaphor, he said that just as the rainbow results from genuine unities (the water droplets), so do the bodies of the natural world result from genuine simple substance. Thus, this relation between the phenomena and the underlying simple substances prevents the phenomena from becoming merely imaginary (as they would be in the philosophy of George Berkley), insofar as they are grounded in the simple substances and therefore become well-founded phenomena.

According to Leibniz, substances are not only essentially unities, but also active, saying in the opening line of his Principles of Nature and Grace, "A Substance is a being capable of action". Because the essence of a Cartesian corporeal substance is extension, said Cartesian corporeal substance cannot itself be a source of activity. Leibniz holds to this principle because, on the one hand, he adheres to the classical/Scholastic idea that actions pertain to suppositions: only something that can be the subject of predication can be active, and only true unities can be geunine subjects of predication (and not mere phenomena). On the other hand, Leibniz hols to this principle becuse he believes that something is active if and only if its activity arises spontaneously from within itself, in accordance with the consequences he drew from his logical notion of substance and doctrine of marks and traces. As Leibniz believed that only minds (or things which are mind-like in the significant ways) can originate and alter their modifications, for this reason he understood the individual simple substances as being mind-like, implictly rejecting the Cartesian mind/body duality by denying the substantiality of material bodies at the discussion's outset.

By saying that substances are essentially active, Leibniz means that the very substance of things consists in a force for acting and being acted upon; each simple substance is endowed with primitive active and passive powers. The primitive active force is understood as the law of the series of the simple substance through time, assing from perception to perception through their own internal strivings by the law of their nature, at the same time harmonizing with one another as they represent the same phenomena of the universe in different ways, a harmonization which must necessarily arise from their common cause. Because the simple substances are thought of like primitive minds, their modifications are considered as representations or perceptions and the activity of the simple substance is thought of as relating to the succession of its perceptions. Each substance may be thought of as having a unique series of perceptions programmed by God to play in harmony with those of all other substances, with the internal tendency of a substance to move from perception to perception being its active force (what Leibniz calls its appetite or appetition).

Pre-Established Harmony

Leibniz argues that God created the world so perfectly that each substance acts according to its own law of unfolding and is at the same time in perfect harmony with all other substances, and that furthermore the mind has a distinct point of view of the world by virtue of its being the center of some mass (one's body) and that the law of the unfolding of the mind is in accord with the laws of the corporeal machine by divine design. This was stated succintly in his 1695 essay A New System of Nature, where he presented what was effectively a five-step argument for his doctrine of pre-established harmony (P1) "There is no real influence of one created substance on another."

(P2) "God originally created the soul (and any other real unity) in such a way that everything must arise for it from its own depths [fonds], through perfect spontaneity relative to itself, and yet with a perfect conformity relative to external things."

(P3) "This is what makes every substance represent the universe exactly and in its own way, from a certain point of view, and makes the perceptions or expressions of external things occur in the soul at a given time, in virtue of its own laws, as if in a world apart, and as if there existed only God and itself."

(P3) "The organized mass, in which the point of view of the soul lies, being expressed more closely by the sole, is in turn ready to act by itself, following the laws of the corporeal machine, at the moment when the soul wills it to act, without disturbing the laws of the other - the spirits and blood then having exactly the motions that they need to respond to the passions and perceptions of the soul."

(C1) "It is this mutual relation, regulated in advance in each substance of the universe, which produces what we call their communication, and which alone brings about the union of soul and body."

When speaking in metaphysical rigor, Leibniz would not ordinarily allow for the underlying premise of Cartesian dualism that body is a substance, as he appears to above; however, in order to deliver his views to those who understood the world in terms of vulgar Cartesian metaphysics, he expressed his position by saying that the mind and body can be said to have a union insofar as the mind and body follow their own seperate laws and the two are in perfect harmony. While Leibniz did not believe that the body and mind were two co-substantial connected through the pineal gland as Descartes suggested, he did believe that the perceptions and appetitions of the soul would arise spontaenously from its own depths to correspond precisely with the actions of the particular body to which it was attached as well as being in perfect conformity with all other substances of the world.

The crucial idea here is that the body follows its own laws and the mind its own laws, without there being any true influence between the two. In Leibniz's view, the mind and body therefore constitute two seperate worlds, unified solely by the correspondance of their actions and perceptions. Furthermore, distinct means of explaining the world may be applied to each of these seperate realms according to Leibniz: we may explain thing in accordance with the final causes of the mind or in accordance with the efficient causes of the body/bodies in general.

Leibniz's account of the pre-established harmony of mind and body is therefore reflective of the more general position taken in his metaphysics, on the existence of parallel modes of explanation for any given occurence. Leibniz believed that the laws of the mind/soul were telelogical, which is to say that he believed that the mind operated for particular ends and that therefore its actions are explicable in terms of final causes; likewise, the actions of the body are purely instances of matter in motion according to mechanical philosophy, and thus are explained in terms of efficient causes. Therefore, in Leibniz's system, the motions of external bodies may be treated as if they acted with no souls, and the internal decisions of one's soul may be treated as if one acted with no bodies, and furthermore when considered together as whole both act as if each had influenced the other, will never be at odds or in discord with one another. 

See Also

Philosophical Critique of Leibniz's System