THE MARCH 1694 edition of Acta Eruditorum included a short article by Leibniz titled “On the Correction of First Philosophy and the Notion of Substance.”1 In it Leibniz complains about an emerging trend of obscurity in metaphysics, attributing it in part to a widening rift between work in metaphysics and work in mathematics. In July of that year Leibniz wrote to Jacques-Bénigne Bossuet, including with the letter a copy of a manuscript titled “Reflections on the Advancement of the True Metaphysics and Particularly on the Nature of Substance Explained by Force.”2 This was an expanded version of the Acta Eruditorum article. As in the article, Leibniz complains about the growing rift between metaphysics and mathematics. His remedy for philosophers is implicit though clear: the growing obscurity in metaphysics could be wiped clean by the co-opting of mathematics. By November of that same year, he looks to have adopted his remedy, writing to the Marquis de l’Hospital, “My metaphysics is all mathematics, so to speak, or it can become so” (1846–60, GM 2:255–62).3 This suggests that Leibniz believed that he had been able to wipe clean any obscurity that may have existed with respect to the most important components of his metaphysics, by clearly articulating them by way of certain concepts taken from mathematics.
In both the Acta Eruditorum article and the manuscript sent to Bossuet, Leibniz claims that the centerpiece of his new philosophical system is his notion (notione) of a substance. For not only does it yield “important truths about God, the soul, and the nature of body” (Leibniz 1998 WF, 141), but it makes a connection between the concepts of substance and force, a connection that turns out to be of great importance to his physics. One of his earliest attempts at discussing the notion of a substance in light of certain mathematical concepts appears in a 1686 manuscript not published during his lifetime, but published later (in 1846) and then officially titled Discourse on Metaphysics.4 So, he had been hard at work for some time (at least eight years) before the Acta Eruditorum article using concepts taken from mathematics to more clearly formulate certain components of his metaphysics. A look at other texts, especially those written around the time of the Acta Eruditorum article, so around 1695, show that after the Discourse he had made advances on his use of mathematics as a means of making clearer his metaphysics, which in turn, as emphasized in this chapter, helped to clarify his physics. Specifically, the advances can be found in the New System and the Specimen Dynamicum. For lack of a better term, I shall call this use of mathematics in metaphysics and physics the “mathematization” of these two sciences. I recognize that this is not the only way this term can be used. I propose to look carefully at the mathematization as it emerges in Leibniz’s work, focusing specifically on how it clarifies his metaphysical concepts of order and harmony, concepts vital to the physics.
The Discourse is among the earliest texts in which we find Leibniz discussing the notion of a substance and what he calls universal order (l’ordre universel) in light of certain concepts taken from mathematics. The universal order, he says, is regulated by the most general of God’s laws (le plus generale des loix de Dieu), which, he is clear to assert, are exceptionless (sans exception; Discourse, §7; Leibniz 1879 [GP] 4:432). Like everything in the cosmos, the notions of individual substances, which we shall see are themselves laws, are also subject to the most general of God’s laws. Although the particular examples taken from mathematics do not help make clearer the sense in which this order is universal, they do help to make clearer Leibniz’s metaphysical conception of order, on which the notion of an individual substance is built. About order he writes:
Suppose, for example, that someone puts a number of completely haphazard points on paper, as do people who practice the ridiculous art of geomancy. I say that it is possible to find a geometrical line whose notion is constant and uniform according to a certain rule, such that the line passes through all the points, and in the same order as they were drawn. And if someone drew a continuous line which is sometimes straight, sometimes follows a circle, and sometimes of some other kind, it would be possible to find a notion (notion) or rule (regle) or equation (equation) common to all the points on this line in virtue of which these same changes would occur. (Discourse §6; GP 4 431)
Order is inherent in the cosmos. “For regards the universal order,” he says, “everything conforms to it. So much is this true that not only does nothing happen in the world which is absolutely irregular, but also that we can’t even imagine such a thing” (ibid.). Even in the case where points are drawn haphazardly on a piece of paper, an equation nevertheless could be constructed that would express a rule that would describe a line that not only connects the points but connects them in exactly the order in which they were drawn. Of course, the line represents only an order discoverable in the data (here, the data are the points), its notion or equation understood as that which expresses that order as represented by that line.
The principal aim of the discussion of the metaphysics of order is to make intelligible the concept of order, to make clear its very possibility. This concept is in turn important to the physics, since one of the aims of the physics is to provide an account of the order of things (as opposed to just an order). The order dealt with in the physics presupposes the metaphysics of order. About his own use of mathematics in the Discourse, Leibniz writes, “I make use of these comparisons in order to sketch some imperfect picture of the divine wisdom, and to say something which might at least raise our minds to some sort of conception of what cannot be adequately expressed” (GP 4 431). In a letter to the Count Ernst von Hessen-Rheinfels, which dealt specifically with issues in the Discourse, Leibniz says that when conceiving the notions of individual substances (the notions understood as “final species”), in line with Aquinas, we should not conceive them “physically, but metaphysically or mathematically” (GP 2 131).5 These statements align with what he will say later in the Acta Eruditorum article, the manuscript sent to Bossuet, the New System, the Specimen Dynamicum, and the letter to l’Hospital.
First let us consider some specific instances in which he appeals to mathematics in making clearer certain metaphysical concepts. As noted, the mathematical concepts of equation and geometrical line were applied in section 6 of the Discourse, where he mentions the order discoverable among the points haphazardly drawn on a piece of paper. These mathematical concepts are again used in section 8, in the discussion of the notion of Alexander the Great.6 In a 1690 letter to Arnauld, we get a slightly different picture. There, Leibniz says of each substance that it “contains in its nature the law (legem) of the continuous progression (seriei) of its own workings and all that has happened to it and all that will happen to it” (GP 2 136).7 Scholars have referred to this law, the law that determines the series of changes in an individual substance over time, as the “law-of-the-series.”8 Leibniz in fact likens the law-of-the-series to what occurs in mathematics, where a rule or equation determines a series in numbers (Cover and O’Leary-Hawthorne 1999).9 We have, then, two competing mathematical “pictures” of an individual substance and the notion that determines (or expresses) it. The first is the picture of a geometrical line and the equation that determines (or expresses) it; the second is the picture of a numerical series and the equation that determines (or expresses) it. They are related but distinct pictures. To be sure, both are metaphors in this context, but they are importantly instrumental in Leibniz’s attempts at making clearer his metaphysics. They are part of what I earlier called the mathematization of his metaphysics. The first picture is again used in section 13, where this time Leibniz offers up a circle in his conception of the notion of Caesar (Discourse, §13; GP 4 437). The second picture is used in section 30. Leibniz says there that God “continually conserves and continually produces our being in such a way that our thoughts occur spontaneously and freely in the order laid down by the notion of our individual substance” (Discourse, §30; GP 4 454). Predicate-talk has been replaced with thought-talk, the notion of an individual substance now cast as that which orders a substance’s series of thoughts. And, in yet another letter to Arnauld, Leibniz puts both mathematical pictures to use, casting the notion as that which orders the unfolding sequence or series of events that constitutes the individual substance while casting this substance’s duration over time, the unfolding of that history of events, as a geometrical line (GP 2 43).10 Although much can be said about the second picture—of individual substance as numerical series11—I shall focus on the first, which depicts an individual substance, and the notion that determines it, in terms of a geometrical line and the equation that organizes items to “produce” it.
EQUATIONS AND ORDER
In an early manuscript, dated July 11, 1677—so almost a decade before the Discourse—Leibniz instructs us on how to construct an equation of a geometrical line.12 The construction is not the point of his discussion, but is required for the more sophisticated mathematical analysis to come. For our purpose we need to focus only on the equation’s construction. First, he constructs a curve DC on paper, and then constructs, relative to DC, two straight lines, AS, which he calls “y,” and AB, perpendicular to AS, which he calls “x.”13 In other letters he will refer to AS as the ordinate and to AB as the abscissa.
The equation of DC, he says, is constructed by thinking of it as expressing a relation between AB and AS. In other words, the equation of DC will produce ordered pairs <xn, yn>, which will be points located on DC, where xn is located on AB and yn is located on AS. Thus, referring now to Figure 1, the equation of DC will produce the ordered pairs <x1, y1>, <x2, y2>, and so on, which, using AB and AS as referents, are points lying on DC. There are a few philosophical issues that should be brought to light, so let me pause briefly to do that, after which I shall return to the specifics concerning equations and geometrical lines.
Figure 1. Algebraic construction of DC from AS and AB.
In 1695, Leibniz tells Foucher, in response to Foucher’s remarks concerning the New System, that there is an important distinction between ideal things such as lines and points, and real or actual things such as corporeal substances (bodies), specifically offering the example of a sheep’s body. One difference between the two kinds of entity is that the sheep’s body, for instance, is a “concrete thing” or “a mass,” and so is an aggregate of an infinity of bodies (though in being a unified living thing is an infinity of bodies organized by the notion of this specific animal, and is presumably a genuine corporeal substance), where its “parts,” those bodies constituting the aggregate, are prior to the whole. By contrast, the geometrical line, an ideal thing, is not an aggregate of its “parts.” Here, the “parts” that he has in mind are in fact points. In speaking about how others have confused ideal and real things, Leibniz says that they have mistakenly thought that “lines are made up out of points” (WF 185). So, the “parts” here are not line segments but are, as he puts it, “the primary elements in ideal things,” or points. For ideal things, the whole is prior to its parts or its elements.14 Even though a line is not an aggregate of points like the sheep’s body is an aggregate of smaller bodies, we can nevertheless conceive a line in terms of “relations which involve eternal truths” (WF 185). If so, what are the relata? According to the above account, DC can be understood as a relation of at least two other lines, AB and AS. Specifically, what the equation of DC shows is precisely how each point xn on AB is related to a point yn on AS. Now, there are an infinite number of possible points or “locations” xn on AB. Likewise for AS—there are an infinite number of possible points or “locations” yn on AS. Despite this, the equation of DC can take any xn on AB and pair it with a unique yn on AS. Here is a sense, then, in which we can understand how it is that an organizing principle orders or organizes infinitely many elements or, as he calls them in the remark to Foucher, “primary elements,” which results in the “production” of a geometrical line. Likewise, I think, the notion of an individual substance S can be understood as ordering or as organizing infinitely many simple substances, which results in the “production” of the individual substance S over time.
This early metaphysical view of an individual substance and the notion responsible for organizing it is, certainly by 1695, extended in some form to all bodies. In the New System, for example, Leibniz makes clear the difference between the souls or forms of a “superior order” and those “sunk in matter which,” he says, “in my view are to be found everywhere” (WF 146).15 The former are those souls that express the notions of individual substances, the latter are simpler in some sense and of an inferior order. Even so, the latter are simple substances, or what in the New System he refers to as “real and animated point(s)” (Leibniz 1695, 145) and later as “metaphysical points” (149). In the Specimen Dynamicum, he casts this animation, or the activity of any informed material locus (i.e., animated or metaphysical point), in terms of conatus (WF 154–79; GM 6 234–54).16 “Since only force and its resultant effort exist at any moment,” he says, “and since every effort tends in a straight line, it follows that all motion is rectilinear or composed of rectilinear motions” (WF 173; GM 6 252) The idea, I think, is that each metaphysical point expresses its own motion, where its motion is rectilinear. This is one sense in which metaphysical points differ from mathematical points. Such motions are smaller than can be measured. In fact, they are smaller than can be perceived.17 Yet out of them arises much of the motion we observe in the cosmos. Though not clear, perhaps these will become the petites perceptions of the New Essays (1703–5)—simple, active, and yet insensible (Leibniz 1991, 49–67; GP 5 41–61).18
Even as late as in the Principles of Nature and Grace (1714), Leibniz writes:
In nature, everything is full. There are simple substances everywhere, genuinely separated one from another by their own actions, which continually change their relationships. Every simple substance, or individual monad, which forms the centre of a composite substance (an animal, for example) and the principle of its unity, is surrounded by a mass made up of an infinity of other monads which constitute the body of that central monad; and in accordance with the ways in which that body is affected, the central monad represents, as in a kind of centre, things which are outside it. This body is organic, when it forms a kind of natural automaton or machine, which is a machine not only in its entirety, but also in its smallest noticeable parts. Because of the plenitude of the world everything is linked, and every body acts to a greater or lesser extent on every other body in proportion to distance, and is affected by it in return. It therefore follows that every monad is a living mirror, or a mirror endowed with internal activity, representing the universe in accordance with its own point of view, and as orderly as the universe itself.19 (WF 258–66; GP 6 598–606)
A “central” monad is the one responsible for organizing a “surrounding mass” of an infinity of monads, this central monad functioning as a principle of unity. “Each monad,” he says, “together with its own body, makes up a living substance. Thus not only is there life everywhere, together with limbs or organs, but there are infinite levels of life among monads, some of which are dominant over others to greater or lesser extents” (WF 260). This, I think, harkens back to the notion of an individual substance as introduced in the Discourse. The dominant or central monad expresses itself as the organizing principle of an individual substance. And although I cannot argue for it here, allow me to at least suggest that Leibniz seems to believe, certainly by the time he writes the Specimen Dynamicum, that this is so whether that substance is Alexander the Great, a red blood cell, the sun, or a piece of iron.
There appear to be earlier versions of the idea of the central monad and the mass of monads surrounding it, as it was introduced in the Principles of Nature and Grace. One such version can be found in the Specimen Dynamicum. This earlier version does not make anything of the distinction between the living body of Alexander, say, and the corpse of Alexander. Rather, it treats of bodies generally, or what Leibniz will refer to as corporeal substances. This is, I think, compatible with the view in the Discourse.
In a 1686 letter to Arnauld, Leibniz had said that the human body “or the corpse, considered in isolation from the soul, can only improperly be called a substance, like a machine, or a heap of stones, which are only beings by aggregation” (WF 117).20 So, postmortem, the corpse of Alexander, supposing that “it” is no longer organized by the notion of Alexander, is now a heap or aggregate of bodies. This aggregate is not an individual substance, for it is now simply a heap of bodies (or corporeal substances) with nothing functionally relating or organizing them into a unified being with a singular aim. When the notion of Alexander did organize these bodies, it organized them so that every body (that now constitutes the heap) was then directed by the notion toward some specific end and no other. This body, the ante mortem “living” body of Alexander, was an individual substance—it was Alexander the Great.21 And, it may even have counted as a genuine corporeal substance. This is suggested when Leibniz writes, “if there are corporeal substances, man is not the only one” (WF 147; GP 4 474–75). He has philosophical reasons for thinking that bodies other than human bodies are corporeal substances. During the period he writes the letter to Arnauld (1686), Leibniz holds that a soul is a purely active being.22 Matter is a purely passive medium through which a soul acts. A body, then, is the result of a soul’s acting through matter, and in this sense, as a union, can be understood to be what he calls a corporeal substance. If this is right, then it would seem that all bodies must be understood as being animate or organic. So, although the corpse postmortem is simply an aggregate of an infinite number of bodies, given that each member of the aggregate is a body, then each is “governed” by some organizing principle, which if not a soul of the superior order is a soul of the inferior order, the kind of soul that God had apparently “sunk” into matter.
In the Specimen Dynamicum, Leibniz tells us that a body will possess two kinds of active force, namely primitive active force and derivative active force. Primitive active force, he says, “is inherent in all corporeal substances as such.” It is “none other than the first entelechy—[which] corresponds to the soul or substantial form.” Derivative active force “is as it were the limitation of primitive force brought about by the collision of bodies with each other” (WF 155–56; GM 6 234–54). This latter kind of force is “the force by which bodies actually act and are acted upon by each other” (WF 157). If we think of the primitive active force as the corporeal substance’s internal principle of organization, we might in turn think of the derivative active force in terms of how this organized body relates to or “harmonizes with” all other bodies. The derivative force, in other words, requires us to consider a body as it relates to all others. Here are a few other details. The velocity (velocitatem) of a body, he says, which in this context seems to be the speed of a body (i.e., the body’s change of place over time), “taken together with direction is called conatus, while impetus is the product of the mass of a body and its velocity (speed)” (WF 157). Here, we begin to see an application of mathematics that is not metaphorical. Impetus is the product of mass and speed (where, if by velocitatem he means speed in a direction—though it is not clear that he means this—then impetus is the product of mass and what we would today recognize as velocity).
Leibniz introduces a number of distinct though related conceptions of force—dead force, living force, respective force, individual force, total force, partial force, directive or common force, to name several. Here, I shall focus on only a few. First, consider two bodies, A and B, where both are in motion. Even though A and B can be understood as bodies themselves, each considered in itself a single body, Leibniz also allows them to be understood as the aggregates of smaller bodies (and each of those smaller bodies the aggregates of even smaller bodies, and so on ad infinitum). Call those bodies constituting A, a1, a2,… an, and those constituting B, b1, b2,…, bm. Respective or individual force, he says, is the force by which the bodies constituting A, for example, namely a1, a2,…, an, can act on one another; likewise, it is the force by which the bodies constituting B, namely b1, b2,…, bm, can act on one another. The interactions occur internal to the aggregates A and B, respectively. By contrast, directive or common force is that force by which the aggregate itself, namely A, can act on something else, for example on B; and it is that force by which the aggregate itself, namely B, can act on something else, for example on A (WF 158–59). Clearly, when we appeal to directive or common force, we treat A and B as singular, or we might even say unified, entities, where no reference to constituent bodies is made. (Of course, A and B could be constituent bodies of some larger body, in which case, understood as constituents the force by which they act on one another will be individual force. But I will ignore this here.) This is consistent with what Leibniz says, for example, in a response to Abbé Gouye, where he notes that depending on what we choose to explain, we can regard the earth, for example, as a point, as it stands to its orbit around the sun, or a ball held in one’s hand as a point, as it stands, say, to the circumference of the earth, knowing full well that they are spheres and not points.23 Likewise, I think, in the Specimen Dynamicum, Leibniz allows himself a way to treat A and B as points, or rather, as points in motion, where the motions of A and B are represented as geometrical lines.
He explains this within the context of introducing the idea of a body’s center of gravity, where this center can be conceived as a (metaphysical) point (WF 173). So, when we think of A as a singular body in motion, we conceive of A in terms of its center of gravity. This comes within the context of introducing two kinds of motion, which not coincidentally are related to the two kinds of force introduced earlier, individual force and common force. Consider again bodies A and B. Individual motion, he says, is the individual motions of the bodies constituting A, namely a1, a2,…, an; and, if considering B, the individual motions of the bodies constituting B, namely b1, b2,…, bm. As was the case with individual force, individual motion occurs internal to the aggregates A and B, respectively. Common motion is the motion of A or the motion of B, each considered as a center of gravity, these centers, as I said, being understood as (metaphysical) points. Now, we know that the notion of A, for example, will be an organizing principle, the central monad to use the language of the Principles of Nature and Grace, that organizes an infinity of a “surrounding mass” of monads. This surrounding mass will no doubt include bodies a1, a2,…, an. Of course, the same holds for the notion of B and b1, b2,…, bm. It is worth noting that the notion of a1, for example, also has a center of gravity, and each body constituting it also has its own center of gravity, and so on ad infinitum. There is much to juggle here. So, how are we to conceive all of this? It is one thing to understand the notion of A as an equation that determines a geometrical line, where the line represents A’s motion, but it is quite another thing to understand how the notion of A works to organize an infinity of rectilinear motions, which include the motions of a1, a2,…, an, and ultimately the distinct conata of an infinity of metaphysical points.
With this sort of challenge in mind, I think, Leibniz develops a mathematical procedure that in fact allows him to solve a system of equations, which makes intelligible a harmony among distinct rectilinear motions, so that the latter can be shown to converge on a singular point, “forming” a body’s center of gravity. The procedure deals with how to find what we today call a determinant. This will have wider application, for it will also help to make Leibniz’s conception of universal order clearer. But I shall focus only on how the determinant makes clearer how the bodies constituting A, for example, might be conceived as being ordered or organized to form A’s center of gravity.
MATRICES AND HARMONY
Leibniz’s development of the determinant is, as far as I am aware, found in only a few texts. The first is a 1693 letter to l’Hospital, the second is an untitled manuscript written sometime before 1693 (GM 2 238–40).24 He does not use the word “determinant.” The term came later. Leibniz begins the description of his new procedure by telling l’Hospital how to understand his new notation, starting with a system of linear equations:
10 + 11x + 12y = 0
20 + 21x + 22y = 0
30 + 31x + 32y = 0
Here, he has set each equal to zero. We are given three equations with two unknowns. Today, mathematicians would say that this system is overdetermined. This means that the number of equations is greater than the number of variables (unknowns). The numbers here are placeholders, and so are not really numbers, but are, as he calls them, pseudo-numbers (nombre feint; S 268; GM 2 239).25 Each is a two-digit pseudo-number, each digit representing important information. For instance, the first digit tells us which linear equation a term belongs to. So, the first digit 1 in the pseudo-numbers 10, 11, and 12 tells us that these terms belong to the first linear equation; the first digit 2 in the pseudo-numbers 20, 21, and 22 tells us that these terms belong to the second linear equation, and so on. The second digit tells us which variable, if any, the term is the coefficient of. So, the second digit 0 in the pseudo-number 10 tells us that this term is the coefficient of no variable; the second digit 1 in the pseudo-number 11 tells us that this term is the coefficient of the x variable; the second digit 2 in the pseudo-number 22 tells us that this term is the coefficient of the y variable. Thus, the pseudo-number 32 tells us that this term belongs to the third linear equation and is the coefficient of the y variable. To emphasize this visually, later in the letter Leibniz writes the first digit larger than the second, the second digit written to look a bit like a subscript. So, 10, 11, 12 become 10, 11, 12.
About the pseudo-numbers, he is clear to say that since they are not really numbers, when he instructs us to multiply 10 and 22, for example, which he writes “10.22,” we should not multiply these as though they were numbers. So, we should not arrive at 220 (ibid.).26 Rather, 10 and 22 are placeholders, 10 standing for the term in the first linear equation that is coefficient of no variable, 22 standing for the term in the second linear equation that is coefficient of the y variable. So, supposing that 3 is in the 10-place and 5 in the 22-place, then 10.22 tells us to multiply 3 and 5. Here, 3 and 5 are numbers. So, the product is 15. Now, the procedure is geared to eliminate and subsequently reduce the number of variables. Ultimately, “the final equation [is] freed from the two unknowns that we wished to eliminate” (S 269; GM 2 239–40).27
Today we do not usually call the terms 10, 20, and 30, that is, terms that are the coefficients of no variable, coefficients. They are referred to as constants. But, Leibniz refers to them as coefficients. Although he appears to be laying out a single procedure, there are, I think, several related procedures. The first can be understood as a method of eliminating unknowns. Working the procedure on two of the above linear equations, we can make out a rule that yields the same results as Cramer’s rule.28 Gabriel Cramer (1704–52) was a student of Johann Bernoulli, and was no stranger to Leibniz’s work. The rule appears in his Introduction à l’analyse des lignes courbes algébraique (1750).29 Let’s look at what Leibniz says, considering two of the above linear equations:
10 + 11x + 12y = 0
20 + 21x + 22y = 0
First, let’s move the constants (coefficients with no variables) over to the right side of their respective equations. So:
11x + 12y = −10
21x + 22y = −20
Leibniz instructs us to multiply the first equation by the y coefficient of the second equation (so, we multiply the terms of the first equation by 22) and multiply the second equation by the negative of the coefficient of the y variable of the first equation (so, we multiply the terms of the second equation by −12) (S 268; GM 2 239).30 So:
11x.22 + 12y.22 = −10.22
−12.21x + (−12.22y) = −12.–20
We can make this more manageable by cleaning up the terms:
11.22x + 12.22y = −10.22
−12.21x − 12.22y = 12.20
Next, we add the two equations together. (12.22y) and (–12.22y) cancel each other out, leaving:
11.22x − 12.21x = 12.20 − 10.22
Now we solve for x:
This quotient is precisely what Cramer’s rule yields.
The rule requires that we first construct the coefficient matrix. This is a matrix containing only those coefficients attached to variables. Looking back at our original two linear equations, these are 11, 12, 21, and 22, the coefficients of the variables x and y. So:
Following Leibniz with what I believe is a slightly different procedure (to be considered shortly), we cross-multiply these terms and take the difference. Specifically, we cross-multiply 11 and 22 (so, 11.22) and cross-multiply 12 and 21 (so, 12.21) and subtract the latter from the former. 11.22 − 12.22 is the determinant of this coefficient matrix. Now, to compute the x value, we substitute in the coefficient matrix the x coefficients with the constants, which are Leibniz’s coefficients attached to no variables. In the two linear equations, these are −10 and −20. So:
We now do the same cross-multiplication and take the difference: −10.22 − (−12.20), which is −10.22 + 12.20, or 12.20 − 10.22. This is the determinant of this new matrix. We compute x by dividing the determinant of this new matrix by the determinant of our original coefficient matrix:
which is the quotient that Leibniz’s procedure produces. To be crystal clear, I will apply this to a real case. Consider this system of linear equations.
−6 + 2x + 3y = 0
−15 + 4x + 9y = 0
Now, to make this simpler, following what we did above, move the constants over to the right-hand side of their respective equations. So:
2x + 3y = 6
4x + 9y = 15
We can construct the coefficient matrix (which, recall, includes only those terms that are coefficients of variables) and apply Leibniz’s cross-multiplication procedure.
(2)(9) − (3)(4) = 18 − 12 = 6
The determinant of the coefficient matrix is 6. Now, let’s substitute the constants 6 and 15 for the x coefficients 2 and 4 in the coefficient matrix:
Cross-multiply and take the difference:
(6)(9) − (3)(15) = 54 − 45 = 9
So, the determinant of this new matrix is 9. Now, we compute x by dividing the determinant of this new matrix by the determinant of the original coefficient matrix.
The procedure for computing y is similar: substitute the constants for the y coefficients of the original coefficient matrix, calculate the determinant of this new matrix (the determinant is 6), and divide this determinant by the determinant of the original matrix (which is 6). Here, y = 6/6 or 1. The solution to this system of equations is < 3/2 , 1 >, which tells us that the lines of this system intersect at this and no other point.
A second procedure allows Leibniz to deal with what I earlier noted was an overdetermined system (more equations than unknowns). Recall Leibniz’s original three linear equations. Remove the variables and addition signs (following Leibniz, I now adopt his alternate notation. So, 10 is now 10, 11 is now 11, and so on.):
10 11 12
20 21 22
30 31 32
This is an ancestor of a 3 × 3 matrix. Leibniz does not call it this, of course. It must be again stressed before getting into the details of the procedure that 10, 20, 30, which we today call constants, are included in this matrix. This is acceptable. But usually when they are included along with coefficients proper (terms that are coefficients of variables) mathematicians refer to the matrix as an augmented matrix. Again, this is a distinction that Leibniz does not seem to recognize. He treats the constants, as I said, as coefficients.31
Here is how the procedure goes. Begin with the matrix-like structure:
10 11 12
20 21 22 = 0
30 31 32
A study of the outcome of his procedure suggests that we should multiply diagonally in both directions. Sum the products when moving right, subtract the products when moving left. Always begin with terms in the first equation. So, moving diagonally (to the right), start with 10, multiply that to 21, and multiply that to 32. So, 10 . 21 . 32. Go to the next term in the first equation, 11, and moving diagonally multiply that to 22, and because we run out of terms, move to the first term of the third equation, 30. So, 11 . 22 . 30. Go to the last term in the first equation, 12, and since we again run out of terms, multiply 12 to the first term of the second equation, 20, moving diagonally multiply that to 31. So, 12 . 20 . 31. Since we got them by moving right, we sum these three products: so, (10 . 21 . 32) + (11 . 22 . 30) + (12 . 20 . 31). So far, so good. But now we must move diagonally in the other direction. So, again begin with 10 and move diagonally this time to the left. Of course, we run out of terms, so move to the last term of the second equation, 22, and multiply by this, then move diagonally and multiply 31. So, 10 . 22 . 31. Continue with the second term of the first equation, 11 (still moving diagonally to the left), multiply that to 20, and since we run out of terms, move to the last term of the third equation, 32. So, 11 . 20 . 32. Begin now with the last term of the first equation, 12, moving diagonally multiply that to 21, and multiply that to 30. Because we got these by moving left, instead of sums we take the difference. Putting this together with the prior sums we get:
(10 . 21 . 32) + (11 . 22 . 30) + (12 . 20 . 31) − (10 . 22 . 31) − (11 . 20 . 32) − (12 . 21 . 30) = 0
Ignoring for the moment that Leibniz is including the constants here, if this were a system of three equations in three unknowns, his procedure in fact produces the determinant of the 3 × 3 matrix. What Leibniz notes, however, is that his procedure will work for “eliminating the unknowns in any number of equations of the first degree, provided that the number of equations exceeds by one the number of unknowns” (S 269; GM 3 5). So, here, insofar as we have three equations with two unknowns, the procedure, he says, should work.32
(10 . 21 . 32) + (11 . 22 . 30) + (12 . 20 . 31) = (10 . 22 . 31) + (11 . 20 . 32) + (12 . 21 . 30)
What Leibniz in fact writes is:
10 . 21 . 32 10 . 22 . 31
11 . 22 . 30 = 11 . 20 . 32
12 . 20 . 31 12 . 21 . 30
Here, he has ordered the left and right-hand sides of the equation using his matrix-like structures, where the first term of the equation on the left-hand side (10 . 21 . 32) is located on top, the second term (11 . 22 . 30) is located underneath that, the third underneath that; likewise the first term of the equation on the right-hand side (10 . 22 . 31) is located on top, the second term is located underneath that, and so on.
What does this tell us? For starters, if the lines determined by the various equations of the system intersect, then there will be a unique, nontrivial, solution. This solution is what Leibniz’s determinant procedure is able to find. By nontrivial it is meant that the equations in the system are not simply equations of the same line, which if so, would entail that there are an infinite number of solutions. And, on the flipside, if all of the lines did not intersect, there would be no solution. What does this mean? Leibniz’s procedure guarantees that if there is a solution it will be a unique <xn, yn> pair that solves each equation in the system. This will be the point at which all of the lines intersect. And this is precisely what we wanted to understand when trying to conceive a body’s center of gravity as a system of rectilinear motions converging on a single point. In terms of body A, and the bodies a1, a2,…, an that constitute it, given that we possessed the linear equation of a1, the linear equation of a2, and so on, and given that they form the center of gravity of A, which is a single (metaphysical) point, Leibniz shows that we can go some way toward understanding, and even calculating, this convergence. To be sure, when A is conceived as a point in motion, the notion of A can be taken to be a linear equation. But conceived as an organized collection of bodies, the notion of A can be taken to be a system of linear equations, where the determinant now helps to make clearer the possibility of A’s center of gravity.
Leibniz employs the above concepts taken from mathematics to make clearer his metaphysical conceptions of order, harmony, and the notion of an individual substance. We saw that he uses the concepts of geometrical line, equation, and what is now called the determinant of a matrix. The geometrical line and equation found a place in his metaphysics, the two used to make clearer his conceptions of order, individual substance, and the notion of a substance, and they also look to have found a place in his physics, the two used to represent the motion of a body. The determinant also looks to have found a place in his metaphysics, though admittedly not as prominently as the line and equation, this apparatus used to make clearer his conception of harmony among individual substances. And it also looks to have found a place in his physics, though again admittedly not as prominently as the line and equation, used to understand how an aggregate of moving bodies might harmonize so as to converge on a single (metaphysical) point, forming a body’s center of gravity. Using mathematics to this end, this is one sense in which Leibniz can be seen as “mathematizing” the sciences of metaphysics and physics.
Leibniz, G. W. 1846–60. Leibnizens Mathematische Scriften
Leibniz, G. W. 1879. Die Philosophischen Schriften von Gottfried Wilhelm Leibniz
Leibniz, G. W. 1976. Philosophical Papers and Letters (followed by volume and page numbers)
Leibniz, G. W. 1959. A Source Book In Mathematics (followed by page numbers)
Leibniz, G. W. 1998. G. W. Leibniz: Philosophical Texts (followed by page numbers)
I am greatly indebted to the editors for their suggestions on how to improve this chapter, especially Benjamin Hill who officially provided comments at the workshop that produced this volume, and, of course, to the participants of the workshop. I also owe thanks to Douglas Marshall for his advice. Last, I thank Roger Ariew, Richard Arthur, Dan Garber, Geoff Gorham, and Doug Jesseph for extended discussions at the workshop that helped me to better understand Leibniz’s view.
1. The original Latin title was: “De prima philosophiae Emendatione, et de Notione Substantiae,” which can be found reprinted in Leibniz (1879, 4:468–70). Hereafter, I shall refer to this collection, the philosophical writings, as “GP,” followed by the volume and page number. I am using an English translation found in Leibniz (1976, 2:432–34).
2. An English translation of this manuscript can be found in Leibniz (1998, 139–42). The original manuscript and correspondence can be found in Bossuet (1912).
3. This letter is No. 1. It is dated November or December 27, 1694. The quote is taken from p. 258: “Ma metaphysique est toute mathematique pour dire ainsi ou la pourroit devenir.”
4. The manuscript is found as an untitled work in GP 4 427–63. The section headings are found in a letter to the Count Ernst von Hessen-Rheinfels, GP 2 12–14. An English translation can be found in Leibniz (1931, 68–72). When not my own, I am using English translations included in WF, 54–93. I am also relying on the translation found in Leibniz (1991).
5. Leibniz to Count Ernst von Hessen-Rheinfels (1687 or 1688). This is letter 24, in Leibniz (1931, 238).
6. Discourse, §8; GP 4 433. Leibniz had in fact sent the Discourse to the Count and to Arnauld, both of whom corresponded with Leibniz and with one another about it.
7. Leibniz to Arnauld, March 23, 1690. This was part of Arnauld’s response to the Discourse. This is letter 26, in Leibniz (1931, 244).
8. See, for example, Cover and O’Leary-Hawthorne (1999). I am indebted to Richard Arthur for bringing this to my attention. I am borrowing the phrase “law-of-the-series” from Cover and O’Leary-Hawthorne.
9. Cover and O’Leary-Hawthorne (1999) quote Leibniz as follows: “the essence of substances consists in … the law of the sequence of changes, as in the nature of the series in numbers” (220). This comes from Leibniz (1923).
10. Leibniz to Arnauld, May 1686. This is letter 8.
11. Cover and O’Leary-Hawthorne (1999) offer an excellent discussion of this picture. It is worth noting, I think, that they actually never take Leibniz up on his use of mathematics in this context. So, their analysis is solely conceptual or logical.
12. The manuscript is included in Leibniz (2005).
13. Here I alter the orientation of his drawing (I spin it 90 degrees counterclockwise) so that AS and AB align with the more familiar y-axis and x-axis as we position them today. Leibniz actually includes two other lines: parallel to AS he puts BC and parallel to AB he puts SC, making a rectangle with the four lines. I only require the simpler drawing to make my point. The inclusion of x1, x2, y1, y2 along with the dashed lines is my own.
14. Here, Leibniz casts points as “extremities” of lines. This is tricky, for it is easy to think that he means that they are simply the “ends” or the “tips,” so to speak, of a finite line segment. But this would be wrong. Rather, in the discussion he speaks of ratios, for example 1/2 and 1/4, where, I take it, he means the following: Assume line PQ. Divide PQ at R. The ratio of PRPQ is a point on PQ. Call this point a1. Now divide PQ at S, where PS ≠ PR. The ratio of PSPQ is a point on PQ. Call this point a2. Since PR ≠ PS, a1 ≠ a2. We construct points, the elements, out of the whole, the line. In this sense, the whole is prior to its “parts.” As I note in the body of the paper, I do not read “parts” as denoting smaller line segments.
15. I cite the English translation in WF 143–52. The quote is from WF 146. GP 4 477–87 has a revised draft. The brackets reflect the additions.
16. From Specimen Dynamicum: An Essay in Dynamics, Showing the Wonderful Laws of Nature Concerning Bodily Forces and Their Interactions, and Tracing Them to Their Causes (1695).
17. For an excellent discussion of the issues surrounding infinitesimals, incomparably small elements, and conatus, which I here suggest by the appeal to metaphysical points, see Jesseph (1998, 6–40).
18. From the preface to New Essays. The remark about insensible perceptions is made in AG 56.
19. From Principles of Nature and Grace Based on Reason (1714), a version of which forms sections 61–62 of the Monadology.
20. Leibniz to Arnauld, November 28/December 8, 1686. This, it seems to me, is aligned with what Aristotle says in the Categories. There, he speaks of a corpse as being a man, but only homonymously so.
21. Alexander’s soul may appear to perish, since it is no longer perceived as organizing bodies existing at a level of normal human perception. This may be what human beings typically call “death.” His notion, however, could continue to operate, but now over bodies that are insensible to human beings. So, Alexander’s visible corpse does not count as evidence of the death (or annihilation) of Alexander’s soul or its activity in the cosmos. About this, see what Leibniz says in the New System (WF 147; GP 4 474–75).
22. See, for example, Mercer (2001). For detailed studies of Leibniz’s views on substance as they evolve over his career, see Leibniz (1994; 2009).
23. I am indebted to Douglas Jesseph’s paper for this passage (1998, 30). There, Jesseph quotes the passage from the Journal, printed in GM 6 95–96.
24. Leibniz to l’Hospital, April 28, 1693. An English translation of a portion of the letter is in Leibniz (1959, 267–69). The untitled manuscript can be found in GM 3 5–6. An English translation of a portion of the manuscript can be found in S (269–70). I rely on Smith’s dating of the untitled manuscript.
25. 1693 letter to l’Hospital.
26. That is, 10.22 ≠ 220. Ibid.
28. I show some of this in Smith (2010, 172–73).
29. Carl Boyer (1985) notes that this rule appears earlier in Colin Maclaurin’s 1748 Treatise of Algebra, published two years before Cramer’s book.
30. Letter to l’Hospital.
31. In Smith (2010, 170–75), I discuss some of this material, though I ignore the present concern over coefficients proper, constants, matrices, and augmented matrices. What I say there is essentially correct, though the procedure I work out is mostly only suggested by what little Leibniz says, though it is consistent with it. I also show how Leibniz’s letter to l’Hospital prefigures Cramer’s rule. This seems to me still to be essentially correct. Finally, since I am confessing things, I overstate in Smith (2010, 175) what Leibniz actually shows in this letter—specifically, I say that he arrives at a particular quotient (which is a quotient that Cramer’s rule yields). While it is true that the procedure Leibniz describes will yield the quotient, he does not actually produce that quotient in the letter.
32. Some of what Leibniz says in these texts suggests that he was toying with what today we think of as cofactors and minors. But this is murky. Even so, given Leibniz’s 3 × 3 matrix we could eliminate the column containing the constants, or coefficients with no variable, by way of an expansion by cofactors. This may be what he means when he says, “Make all combinations of the coefficients of the letters, in such a way that more than one coefficient of the same unknown and of the same equation never appear together. These combinations, which are to be given signs [plus or minus] in accordance with the law which will soon be stated, are placed together, and the result set equal to zero will give an equation lacking all the unknowns” (S 269; GM 3 5). But restrictions on length of chapters in this volume prohibit me from pursuing that here.
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