11

THE GEOMETRICAL METHOD AS A NEW STANDARD OF TRUTH, BASED ON THE MATHEMATIZATION OF NATURE

URSULA GOLDENBAUM

Mais ie n’ay resolu de quitter que la Geometrie abstracte, c’est a dire la recherche des questions qui ne seruent qu’a exercer l’esprit; & ce affin d’auoir d’autant plus de loysir de cultiuer vne autre sorte de Geometrie, qui se propose pour questions l’explication des phainomenes de la nature.

—DESCARTES TO MERSENNE, JULY 27, 1638

THE CONNECTION BETWEEN GEOMETRICAL METHOD AND MATHEMATIZATION OF NATURE

Use of the geometrical method has long been criticized, even before Kant, for being inappropriate in the field of philosophy. There is above all the general reluctance to accept the ponderous method of geometrical demonstration in philosophy. This method is considered to require definitions and demonstrations of propositions,1 rarely commenting and explaining, thus providing little communication with the audience, while eschewing irony and rhetoric altogether. Many philosophers characterized this geometrical method as a mere external means of presentation, without contributing anything philosophical.2 In the case of Spinoza, his use of the geometrical method was even considered as a major hindrance to the reader’s getting into his precious inner doctrine.3

My aim in this chapter is to argue that the geometrical method is not just a form of presentation or demonstration. Rather, this method was seen as a new standard of knowing natural things. It was embraced as a new epistemological approach to the external world, laying the ground for a new type of science and philosophy. The geometrical method of early modern times was no longer just geometry but was clearly connected with the new science of mechanics. It was the wedding of mathematics and the art of mechanics, that is, the art of building machines that brought about Galileo’s new science of mechanics. Turned into a science, mechanics was no longer the art of building machines and “tricking nature.” Rather, the entire world was now considered as a machine, constructed by a divine mechanic.

While this new mechanical science owed much to geometry, it in turn had the greatest impact on geometry and mathematical development in general.4 While Euclid had defined a line by the motion of a point rather accidentally, early modern mathematicians and then Hobbes made the generation of geometrical figures the starting point for understanding geometrical figures through their causes. Hobbes, moreover, used this new explanatory device far beyond mathematics and formulated the general epistemological principle that we can only know what we can generate (OL I, 9; De corpore I, 1, #8). Alan Gabbey (1995) first pointed to the often overlooked achievement of Spinoza who first used the geometrical method to understand the functioning of human beings, especially of their emotions and of the consequences for their morals and politics (142–91). Of course, Spinoza also uses the geometrical method to present his argument by compelling demonstrations. What is of greater significance though, is the fact that he uses this method to lay ground for a scientific ethics: “In this respect the Ethics was more radical than (say) Newton’s Philosophiae naturalis principia mathematica (1687), where at least the mathematica and the philosophia naturalis were both parts of philosophia speculativa” (Gabbey 1995, 147–48).

Spinoza’s entire metaphysics, the notorious parallelism of his philosophical system, rests on the new geometrical method, namely on the conviction that ideas follow one another in the same order and connection as things in the world, causing one another. According to the analytical geometry of Descartes and Viète, the figure of a circle parallels its equation if considered in the framework of the Cartesian coordinates with a defined unit. There the figure of the circle was substantially identical with its equation without any visible similarity of the figure with its equation.5 What gave rise to the geometrical method as a new epistemological standard in philosophy, embraced by all rationalists, was Galileo’s mathematization and thereby mechanization of nature, the wedding of mathematics to mechanics.

These philosophers saw mathematics and the geometrical method as the high road to certainty in human knowledge while raising caution about mere sense experience. The mathematical method could not only provide certainty but also freedom from authorities and from biased interpretations of mere empirical facts: “For there is not one of them,” Hobbes complains about past philosophers, “that begins his ratiocination from the definitions, or explications of the names they are to use; which is a method that hath been used only in geometry, whose conclusions have thereby been made indisputable” (Hobbes 1994, 24; Leviathan v, 7). And Spinoza states that the belief alone “that the judgments of the Gods far surpass man’s grasp … would have caused the truth to be hidden from the human race to eternity, if mathematics, which is concerned not with ends, but only with the essences and properties of figures, had not shown men another standard of truth” (Ethics I App, C 441; emphasis added).

This enthusiasm for the new geometrical method in close connection with the mathematization of nature stirred up resistance. The aversion of theologians—as well as of Christian philosophers such as Henry More, Locke, Kant, and the German idealists—against the geometrical method was not a result of their deep mathematical insights. Rather it was due to their fear of necessitarianism as well as of hubris. They feared that the geometrical method, with its claim to provide knowledge as certain as that of God (adequate ideas), would lead to the claim of human “omniscience.”6 The mathematization of nature would introduce the mathematical necessitarianism into nature and take away God’s will and humans’ free will.7 Fighting Wolffianism, Joachim Lange, the chief pietist theologian, and Valentin Ernst Löscher, the leader of orthodox Lutheran theologians (Löscher 1735, 126–29), both saw the geometrical method, especially the genetic definitions, in clear connection with the mathematization of nature.8 They saw the aim of certainty of human knowledge as delivering divine certainty to human beings. They considered the view that genetic definitions provide the highest standard of truth as a threat to throw theology from the throne in order to install philosophy as the highest knowledge.9 Lange and Löscher both blamed the geometrical method for dismissing the truth of the Christian religion10—because the latter rested on historical knowledge alone, thus lacking adequacy.11

What made theologians so vigilant against this method is just the assumption that we can use mathematical and further mechanical methods to study the inner structure of nature, God’s creation—and that we can come up with a kind of knowledge that is superior to mere sense perception and mere empirical investigation, nay, which is even like that of God himself. In contrast, the Lutheran theologians praise empiricist philosophers and scientists.

As a matter of fact, such fundamental theological suspicion against mathematics as a tool of natural science had been expressed by Catholic theologians before, against Galileo—that is, right from the outset of the new mathematical science of mechanics. In the documentation of the trial one can find, among other listed accusations against Galileo, “to badly state and declare that there is a certain equality, in understanding geometrical things, between the human and divine intellect” (Galilei 1907, 19:326–27).12 Thus, from the very beginning, the enthusiasm to mathematize the science of nature and thereby gain access to the inner structure of nature through discovering mathematics within nature stirred up strong theological resistance.

This critical approach to the mathematization of nature was perfectly in agreement with the tradition of Aristotle and even of Plato (Cassirer 1946, 277–97). With all respect and enthusiasm for mathematics, they would never have thought of nature, that is, of pebbles, mountains, rivers, plants, or animals, as being mathematically structured. These natural things could be known only by observation and classification. In contrast, Galileo, and thereafter Descartes, Spinoza, and others considered nature to be constructed mathematically. Therefore, we needed mathematical science to learn about the functioning of nature. Cassirer emphasizes this difference: “For what does the term ‘science’ mean in Galileo’s system? It never means mere probability, it means necessity. It means no mere aggregate of empirical facts or haphazard observations; it implies a deductive theory. Such a theory must be capable of demonstration; it cannot be based on mere opinion or probability. If it is not possible to attain such a deductive truth about physical phenomena, then Galileo’s scientific ideal, the ideal of modern dynamics, breaks down” (Cassirer 1946, 281). And indeed, the term “necessary demonstrations” as Galileo’s tool of mechanical science occurs again and again in his writings.13

In early modern times, it was first the theologians who defended the view that nature, as God’s creation, is incomprehensible to human beings, knowable only by observation, externally, and by chance. This defensive position was soon taken up by (Christian) philosophers, who tried to escape the mathematization of nature and thereby escape the comprehensibility of nature by mathematics (Goldenbaum, forthcoming, “How Theological Concerns Favor Empiricism over Rationalism”). I see mathematizing nature and the rejection thereof as the root of the well-known opposition of the two philosophical camps of rationalism and empiricism, an opposition that continued into the battles between Wolffianism and Pietism during the eighteenth century in Germany, and which provided the background for Kant’s entrance into philosophy. Kant’s well-known rejection of the geometrical method as inappropriate did not result from Kant’s work on mathematics either (Kant 1998, 630–48). He had argued against the mathematization of nature and the geometrical method in his very first book on the estimation of forces (Goldenbaum, forthcoming) when he barely knew recent mathematics.14 He simply shared the German Lutheran theologians’ fear of necessitarianism and tried to save free will by allowing a mutual influence of body and soul (Goldenbaum, forthcoming).

In addition to the traditional objection against human hubris when claiming a human knowledge of nature like God’s own, there lingered another threat for Christian religion that arose out of the mathematization of nature—namely necessitarianism. What constitutes the great advantage of mathematical cognition, namely necessary knowledge, would allow for necessary knowledge about nature as well—if nature were mathematically structured. Natural processes were necessary and could be comprehended by means of causal connections. As a result, if mathematics were to structure God’s creation it would mean the end of free will, that of God as well as of human beings. Neither would God be capable of acting arbitrarily according to His good will, nor could human beings have a free will to act according to the good or bad. It was these theological concerns that produced a deep suspicion among Christian philosophers against using the geometrical method beyond mathematics. Exemplary of this philosophical suspicion against the mathematization of nature and the geometrical method is the development of Henry More’s relation to Descartes and his philosophy. While More had never been a partisan of Descartes, he admired his work in the beginning and appreciated the special status of the soul in this philosopher’s system. However, from More’s first comments one can grasp his sensitivity regarding Descartes’ emphasis on the geometrical method, on mathematics as the high avenue to knowledge of nature, and on the necessity of the knowledge we are able to obtain in this way (More 1711, esp. 3, 8, 40). But More would turn against Cartesianism as atheism as soon as its necessitarianism became notorious, through Descartes’ partisans, as Alan Gabbey has shown (1982, 173–74). Although More would never accuse Descartes himself of being an atheist, he recognized that the mathematization of nature would inevitably result in necessitarianism, leading to atheism. This became manifest with Hobbes’s Leviathan, Ludewijk Meijer’s Philosophia Scripturae interpres, and ‘Cartesian’ writings, not to mention Spinoza (ibid., 233–50). Of course, the critics of the mathematization of nature well agreed that we can produce mathematics within our minds. But they strongly detested mathematics in nature, outside of ourselves; or, if there were mathematics in nature, they held that we could not have any access to it due to our limited capacities. As a result, John Locke even stated that there could never be such a thing as a science of natural bodies (Locke 1975, 560; Essay IV, 3, #29).

In this chapter I first explain this new geometrical method and its new use in philosophy and science in general, thereby refuting the above-mentioned prejudices against its appropriateness in these fields (see next section). I then focus on the concept of definitions as the cornerstone of the new geometrical method and discuss the concept of adequate ideas, showing how it is closely related to the concept of causal definitions, which was first introduced by Thomas Hobbes. I then describe how the Christian philosopher Leibniz embraced the new geometrical method and the concept of causal definitions that led to his logical containment theory and how he struggled to avoid necessitarianism while retaining the geometrical method and the mathematization of nature. Leibniz was aware that it was the threat of necessitarianism that caused theologians and especially British philosophers to reject the geometrical method in metaphysics. He assured the Cambridge Platonists that, based on his approach, mathematics and the geometrical method would no longer threaten free will and could be successfully applied (GP 3 363–67, 401–3).

WHAT IS THE GEOMETRICAL METHOD?

Although ancient mathematics used the method of deducing demonstrations from axioms and definitions, surprisingly, the term “geometrical method” came into use only during early modern times. Since Zabarella, it has been described as involving two aspects, both belonging to the method, namely the resolutive and compositive methods, also known as the analytic and synthetic methods (Cassirer 1974, 1 136–44). While the former is considered to be helpful for discovery and invention (but difficult to bring under rules), the latter is appreciated for ensuring the certainty of the results due to a complete deduction of propositions; that is, demonstration. Thus it was the synthetic method in particular that provided the compelling force to convince others of the correctness of a solution, while scientists and especially mathematicians did not care much about a gapless deduction if they were working to solve a problem (Breger 2008, 191–92). It was above all the compelling force of the method that made it highly attractive to early modern philosophers and scientists. The familiar anecdote about Hobbes’s wondering about the Pythagorean theorem and being struck by the compelling power of its demonstration may illuminate this enthusiasm (Aubrey 1898, 1 332–33; Hobbes 1994, lxvii). But it was this compelling power of the geometrical demonstration as well that caused theological concerns about the method, and, moreover, about mathematics in general, as soon as this geometrical method was applied beyond pure mathematics.

Usually, when we think of geometrical method today we associate it with what we see when we open a book of Euclid, or (if we are looking for its use in philosophy) what we see in Spinoza’s Ethics. Instead of a coherent flow of text, the lines are broken up into different types of text—definitions, axioms, postulates, propositions, and demonstrations. Although geometrical method is often associated with such complicated mathematical procedures, the great mathematician Pascal broke it down to only two essential rules in his fragment on the geometrical spirit that came down to us as a part of the Logic of Port-Royal: (1) not to employ any term whose meaning is not defined and (2) not to advance any proposition that is not demonstrated by known truths (Pascal 2000, 155–56). These simple rules do not sound as if they cannot be applied to disciplines other than mathematics and it seems difficult to understand what would be wrong with applying these rules to philosophy.

What is interesting from Pascal’s precise and brief formulation is the fact that it is not, as usually believed, demonstration that is at the heart of this method. Nor are axioms essential to the geometrical method as it is often said. This misconception sometimes leads to an inappropriate identification of the geometrical method with the axiomatic method. However, the definitions are essential to the new geometrical method. Every demonstration has to set out from a definition. It is nothing more than a deduction of concepts from the concepts virtually included in the definition at the beginning. Hobbes considers a demonstration as a mere chain of definitions (OL I 252–58; De corpore 3, 20, §6), and Leibniz follows him in that.15 Given this crucial role of definitions for the new geometrical method, it does not come as a surprise that the concept of the definition moved to the center of the long-lasting discussion of geometrical method in the seventeenth and eighteenth centuries.

While the opponents of the geometrical method had to agree that its results did indeed produce necessary conclusions, to dispute the geometrical method they instead questioned the certainty of its very beginnings, of its foundation, that is, of definitions and axioms. Both of these concepts became the subject of hot debate. Their disputed status in terms of certainty served as a bulwark of the critics of the geometrical method (Pascal 2000, 573 [no. 101]). Theologians and other defenders of Christian religion pointed to axioms as undemonstrated assumptions and to definitions as arbitrary human settings lacking the true essence of defined things on which nevertheless all the wonderful demonstrations rested, thus making the entire building of mathematics uncertain in its foundation.16

In principle, the problem with the axioms was easily solved by Hobbes, followed by Leibniz, by simply reducing axioms to demonstrations (OL 1 105–6; De corpore 2, 8, §25; Leibniz A II, 1 281; A VI, 2 480). Both philosophers argued that axioms are indeed assumptions, considered self-evident and therefore accepted by everybody. But as soon as anyone raised a doubt about any axiom it had to be demonstrated from definitions alone. Similar statements can be found in Spinoza.17 Hobbes and Leibniz both showed this in an exemplary way for the famous (and in their days suddenly disputed) axiom that the part is smaller than the whole (Hoffmann 1974, 12–14; Goldenbaum 2008).

The question of definitions—whether they could be formulated at will by human beings and were thus under our control although lacking correspondence to the essences of real things, or if they were something objective albeit incomprehensible—was more problematic. This discussion continued from the seventeenth century well into Kant’s days. The point of disagreement was whether definitions of things, which were not mere products of our minds, but real things, independent of our minds, could be defined by human beings at all.18 The opponents argued that we had to empirically find the real things’ properties and to build nominal definitions of collections of properties of a thing. There was no way to come up with essential definitions as we could provide in mathematics where we could construct the objects of our definitions and thus know their essence. The defenders of the use of the geometrical method beyond geometry emphasized the continuity between objects of pure mathematics (which were considered to be products of human minds) and objects of natural science (which were considered to be products of God’s creation), thus allowing the use of the geometrical method within science and philosophy.19

DEFINITIONS IN THE FRAMEWORK OF THE NEW GEOMETRICAL METHOD

Arnauld and Nicole in their L’art de penser maintained the traditional distinction between nominal and real definition (Arnauld and Nicole 2011, 325–31; L’art de penser 1, 12), rooted in Aristotle’s Organon (Anal. Post. 2, 7–10). While the first was nothing more than words by which we named things, either by convention or by custom, without knowing the essence of a thing the latter would allow us to understand whether the defined thing was real or at least possible in reality. Pascal did not accept any other than nominal (i.e., merely arbitrary definitions).20 Likewise, he also saw axioms as a starting point without being able to secure their certainty, thereby making all our knowledge generally limited.21 Denying our ability to come up with any real definition simply meant we could not know the essence of anything or, put differently, have any objective knowledge about the natural world. Therefore, we had to rely exclusively on experience to know something about the external world. We could observe and look for regular patterns.

Although it is Thomas Hobbes who is often blamed for the doctrine that definitions are arbitrary, he in fact rose to Pascal’s challenge. We have seen already that Hobbes successfully addressed the problem of the axioms by showing they could be demonstrated if needed. But he also developed a new general approach to the other problem—that of real definitions. The way he does this sheds quite some light on how the new geometrical method of early modern time was indeed new—he connects the issue of definitions with Galileo’s new science of mechanics.22 Hobbes considered geometrical figures as produced by mechanical motion. A circle is produced by the motion of one endpoint of a straight line around the other endpoint. Thus Hobbes first introduced this new mechanical approach to definitions systematically into philosophy and demanded causal definitions (or genetic definitions) in philosophy in order to produce necessary conclusions about reality. A definition that includes the mechanical cause of the thing to be defined can serve to deduce all the properties of the thing (OL 1 71–73; De corpore 1, 6, §13).23 To wit, such causal definitions provide the opportunity to deduce any possible property of the circle, even of properties of which we are not yet aware.

But while Hobbes and Pascal both claimed that definitions were arbitrary, Hobbes accepted nominal (i.e., arbitrary) definitions only for “absolute knowledge,” that is, for factual knowledge. Such absolute knowledge, according to Hobbes, is provided only through experience, and does not require knowing the cause of the thing (Hobbes 1994, 35; Leviathan vii, 3). Therefore, in the case of empirical knowledge we are free to name observed sensations and thus can produce arbitrary definitions. But when we can produce the mechanical cause of a thing to be defined we can provide causal definitions. While we cannot draw any necessary but only probable conclusions from nominal definitions, causal definitions guarantee necessary conclusions. This has been noticed in Hobbes scholarship in recent decades (Jesseph 1999, 198–205).

Interestingly, Hobbes’s innovation of causal definitions was then adopted (together with the geometrical method) by all rationalists—by Spinoza (TIE §69–71), by Leibniz (see below), and by Christian Wolff (Cassirer 1974, 2:521–25; Goldenbaum 2011b). With the exception of Spinoza, they hardly used the explicit form of the geometrical method but all their works follow this very method, that is, they begin with definitions and deduce the entire argument from them.24 It should be noticed that all rationalists were mathematicians! Of course, Descartes and Leibniz were geniuses in mathematics whose achievements are still recognized today. However, Spinoza and Wolff well knew the most recent mathematics of their time and could follow the ongoing discussions. And the model for causal definitions is clearly the geometrical construction of figures. But neither Hobbes nor other rationalists stopped there.

The new approach to geometrical method, based on a new concept of causal definitions, was no longer Euclid’s method but went far beyond his project. The most significant difference between the new geometrical method and what ancient geometers did is its extension beyond geometry. The most famous example is of course Spinoza, who wrote a metaphysics or rather an ethics according to this method. Descartes saw all his science as mere mathematics (AT 2 268). But Hobbes had already claimed that there cannot be any science that does not use the geometrical method or draw conclusions from causal definitions.25 While no merely empirical discipline could ever turn into science because no result of these disciplines could aim for certainty, that is, for necessary knowledge, mathematics, optics, mechanics, and—famously—politics could become science because they all started from causal definitions. Hobbes’s surprising inclusion of politics and Spinoza’s treatment of ethics among strict sciences follow precisely the model of the causal definition as suggested for early modern geometry: knowing the mechanical cause of a thing, as the mechanical motion bringing about a circle, leads to certain knowledge of the effect. Knowing the mechanical causes that bring about a commonwealth, we can know the commonwealth, its rules and needs with necessity, with absolute certainty, a priori.26

Of course, all rationalists acknowledged the limits of their reason. They already knew the dilemma famously formulated by Einstein: “How is it possible that mathematics, being a product of human thinking independent of all experience, fits the objects of reality? Can human reason, without experience, explore the properties of real things, by mere thinking? There is a short answer to this, according to my opinion: To the extent that the propositions of mathematics relate to reality, they are not certain, and to the extent they are certain, they do not relate to reality” (Einstein 1921, 3–4). We could not know of external things without experience and experience could not provide us with the essence of things. We were thus forced to give provisional names, that is, nominal definitions, of the things we knew through experience. Nominal definitions were considered to be placeholders for the time being. Hybrids are considered possible. There can be a thing like a commonwealth, for Hobbes, Spinoza, or Wolff, which cannot be known through and through by causal definitions because the biological nature of human beings is still largely unknown. However, one can define human beings for the time being as animals with some use of reason, based on experience. Using these provisional definitions, one can then find merely theoretical explanations using nothing but known terms. In this way, one will be able not only to explain the rules of politics or human behavior but to predict other phenomena sufficiently.

Christian Wolff used this method systematically to reduce the gap between a priori knowledge and experiential knowledge. That is not only true for his experimental physics. When he wrote about methods to increase the growing of grain, he distinguished between facts we know from experience and the causes of some phenomena, which we know with certainty and have under control (reproducing grain) (Wolff 1734; Goldenbaum 2011b). Although we cannot know the essence of the plants yet, we can come to know some causal processes of the plants’ growth and can predict the outcome with a high degree of certainty.

Spinoza used this geometrical method in his theoretical published work on ethics and in his work on politics and biblical hermeneutics. According to Tschirnhaus’s reports to Christian Wolff, Spinoza had developed a method for finding and constantly improving definitions in empirical natural science27 using experiments, starting with arbitrary nominal definitions and increasingly replacing parts of them with causal definitions (Goldenbaum 2011b, 29–41). Tschirnhaus, who was above all a mathematician and engineer (among other things, he invented Meissen porcelain), further developed this method of defining and redefining objects of natural science based on empirical research, as did Christian Wolff. It was their goal to improve the definitions of real things, not only geometrical figures, in such a way that valid conclusions could be drawn from them necessarily, thus extending the realm of the geometrical method far beyond geometry. In their view, we could indeed learn to know such real natural things in the same degree as God, although only to the extent to which we could generate them. Although we could not know natural things thoroughly, we could always try to know some of their properties in a causal way and thus necessarily, or a priori.

All these attempts clearly show that these rationalists used the geometrical method in the most general way to explain not only geometrical figures but as many phenomena of the real world as possible, by finding their causal definitions (using the analytical method). This is held to be true even if we can generate a thing in a way different from the way in which it was actually produced. Whenever a thing is produced and is thus possible, free of contradiction, its essence can be known. These essences (i.e., causal definitions) are connected to one another and have to be compatible, that is, they build a coherent conceptual structure of the world. Although we can only know a small number of particulars in such an a priori manner (because we can generate them), due to their absolute certainty no empirical knowledge can ever contradict them. Thus we can know some eternal and fixed structures, to which all empirical knowledge of particular things must cohere, allowing us to build one coherent structure of the world (although it will always remain incomplete). It is seldom noticed that exactly this position was already held by Galileo: “all these properties [of things in nature] are in effect virtually included in the definitions of all things; and ultimately, through being infinite, are perhaps but one in their essence and in the Divine mind” (Galilei 1967, 104).

ADEQUATE IDEAS AND CAUSAL DEFINITIONS

The mathematician and rationalist Descartes did not speak of causal definitions. But a kind of prehistory of causal definitions can be found in his discussion of adequate ideas with Arnauld, on the basis of Descartes’ Fourth Meditation. The term “adequate ideas” is, of course, more familiar to us from Spinoza and Leibniz, as well as from Wolff. (Hobbes did not use it, perhaps because of his avoidance of traditional scholastic metaphysics in general.) Descartes uses the term cautiously. He states, “if a piece of knowledge is to be adequate it must contain absolutely all the properties which are in the thing which is the object of knowledge” (CSM 2 155; AT 7 220; Meditations, Fourth Set of Replies). Interestingly, adequate ideas have the same capacity as causal definitions, namely the capacity to virtually include all properties that belong to the cognized/defined thing.

It is in this first emergence of adequate ideas in rationalist modern philosophy that we are likewise confronted with the sensitivity of theology regarding adequate ideas. Descartes immediately adds a caveat: “Hence only God can know that he has adequate knowledge of all things. A created intellect, by contrast, though perhaps it may in fact possess adequate knowledge of many things, can never know that he has adequate knowledge unless God grants it a special revelation of the fact” (ibid., emphasis added). Why is the talk about adequate ideas immediately turning to theology? Because having adequate knowledge of things makes us like God—knowing things as well as He does in His omniscience. Descartes was as aware as anyone of the theological concerns regarding Galileo. Therefore, in spite of his enthusiastic statements about the certainty of deduction and intuition (both of which are available to us) in his early writings, especially in the Rules, he has to backpedal and grant that God could have made the world in a way that would be completely incomprehensible to us, in opposition even to what we hold to be mathematically necessary.28 Of course this position caused headaches for Leibniz and other rationalists.

This is not the only reason adequate ideas are, from their first appearance in Descartes’ discussion with Arnauld, an extremely sensitive topic in terms of theology. This discussion about the Fourth Meditation is titled “De vero et falso” and deals with the question of how error arises—although the perfect being, God does not deceive us. Descartes does not ascribe the reason for our error to God or to human reason. Rather, Descartes considers it to be an easy thing to know many things adequately as long as our vis cognoscendi is adequate to the thing to be known, “and this can easily occur” (ibid.). Descartes finds the cause of error, as we all know, in our will. This is where Descartes struggles to argue in favor of the free choice of our will. And this topic of free will is obviously tainted by deep theological concerns.

What is important for my point is rather Descartes’ first concern. In order to know that we have the power to cognize things adequately and to know “that God put nothing in the thing beyond what it [our mind] is aware of”—we would have to know everything. Thus our power of knowing would have to equal the infinite cognizing power of God, which is clearly impossible. Descartes cautiously concludes that we do not need adequate ideas to conceive the real distinction between two subjects, here mind and body. And he suggests that we may be content with “complete” ideas that would give us all the properties of a thing (i.e., as much as an adequate idea) anyway without claiming its adequacy. By this distinction of adequate and complete ideas, the first owned by God alone and the second available to us, Descartes guarantees a limit to what can be known by human beings. In his Rules and then in Discours, Descartes did attribute to human beings a capability of knowing things with certainty by relying on intuition and deduction, or the geometrical method.

Interestingly, Descartes’ cautious distinction between adequate and complete ideas was not upheld by his followers. For Spinoza it is precisely our adequate ideas that provide for our sharing of God’s intellect, allowing for certainty of our knowledge (E II, 37–40s2) and overcoming our lack of freedom (E IV). Adequate ideas will even make us immortal (E V 38–42s). Spinoza defines “adequate idea” as “an idea which, insofar as it is considered in itself, without relation to an object, has all the properties, or intrinsic denominations of a true idea” (E II, d4; C 447). Thus he explicitly denies correspondence of an idea with an external object as a criterion for adequacy and thereby denies the traditional understanding of adequacy in Aristotelian scholastics as correspondence of idea and ideatum.29 For Spinoza, having an adequate idea is to provide the proximate cause of the thing to be known, that is, the idea that causes an idea, or to define a thing by its cause, if considered under the attribute of extension.

In contrast to Descartes, Spinoza holds that we can have such adequate ideas and produce more of them when following the geometrical method and working to obtain increasingly causal definitions (or at least partially causal definitions), using nominal definitions as mere placeholders. Of course, being finite, we can never come even close to God’s intellect. God knows everything adequately and moreover intuitively; however, we can get to know some important adequate ideas, which may then provide a general structure to lead our empirical research in a safe way. That is, because “the fixed and eternal things” (TIE 101; C 41) are so closely connected to the particular things, their knowledge will help us to get a more coherent knowledge of the latter. Thus Spinoza allows human beings to have adequate ideas and even sees these ideas as divine knowledge, which we share with God, clearly deviating from the cautious position of Descartes. When Spinoza discusses inadequate ideas and explains error he even ironically uses the example of free will as an exemplary inadequate idea (E 2 p35sch).

Interestingly, and seldom noticed, this rationalist position is very close to that of Galileo, who claims (just as the theologians complained in the earlier quoted trial file): “I say that as to the truth of the knowledge which is given by mathematical proofs, this is the same that Divine wisdom recognizes” (Galilei 1967, 103; emphasis added). Of course, Galileo admits a difference between divine and human knowledge—a difference consisting in God’s thoroughgoing intuitive knowledge in contrast to human discursive knowledge. But still, he vindicates “a few” intuitive insights to human beings too.30

Even Leibniz, the committed Christian philosopher, accepted that we have the capability to have adequate ideas. And he also agreed that they are the same in us as in God, to the extent that we have them, because they are necessarily true. As Spinoza does, Leibniz connects them with genetic or causal definitions, which necessarily provide truth. It is interesting that in Wolffianism, when it comes to German translations, the term “idea adaequata” is bluntly translated as “complete idea” [vollständiger Begriff] (Sittenlehre 1745), thereby ignoring Descartes’ careful distinction between complete ideas available to human beings and adequate ideas available to God. However, while all rationalists agree that human beings can have a certain number of necessary demonstrations (i.e., a priori knowledge equaling divine knowledge, the latter claim not being shared by Hobbes), this view is moderated by their awareness that such a priori knowledge is very limited in human beings and has to be supplemented by experience.

Leibniz on Causal Definitions, Adequate Ideas, and Necessitarianism

Given the theological concerns with the geometrical method and adequate ideas, it is rather surprising how close the Christian philosopher Leibniz’s positions on these topics came to those of Spinoza and Hobbes, especially in light of the cautious attitude of Descartes about religiously sensitive issues. In his well-known Meditations on Knowledge, Truth and Ideas from the period of the mature Leibniz, the German philosopher introduces a full-fledged schema of different types of ideas. He first aligns with Descartes to distinguish between obscure and clear ideas but then further splits the clear ideas into clear and confused and clear and distinct, thereby adding a new type: clear and confused ideas (L 291–95; A 6, 4, N. 139, 585–86).31 Further, Leibniz divides distinct ideas into inadequate and adequate ideas, and the latter again into symbolic or intuitive ideas. An idea is adequate if everything that has gone into a distinct knowledge of the thing is also known distinctly “or if the analysis has been done to the end.” Thus, when we can provide distinct knowledge of all partial concepts of a concept we can know it adequately. Leibniz cautiously adds that he does not know whether we have any perfect example of adequate ideas within human knowledge but the knowledge of numbers would come close to it.

Adequate ideas—from Descartes via Spinoza to Leibniz—are those that provide a complete knowledge of all the properties of their subject, independent of any knowledge of correspondence. How does Leibniz relate adequate ideas to causal or genetic definitions? He explains this very systematically in a text he did not publish, On Synthesis and Analysis (L 229–34; A 6, 4, N. 129) (the title refers to the two aspects central to the geometrical method, as mentioned above). He begins with the traditional distinction of nominal and real definitions as still taught in the Logic of Port-Royal, but then emphasizes one particular kind of real definition, which displays the reality of things to us, namely causal or genetic definitions. Starting with nominal definitions, the collection of names of properties of a thing known by experience, he defines them as distinct concepts because it is necessary to distinguish and name the single properties of the subject to come up with nominal definitions. Confused ideas though, for which we cannot give single properties although we somehow recognize a thing in its entirety, do not allow yet for any definition. They may be made more (and more) distinct though by analysis, that is, by further distinguishing their parts.

In contrast to such nominal definitions, being a mere listing of properties or, rather, their names, Leibniz then defines real definitions as including the possibility of the defined thing, or freedom from contradiction. His example is—surprise!—the definition of a circle; specifically, Euclid’s definition of a circle as produced by the motion of a straight line in a plane around one of its unmoved endpoints. This definition, clearly a causal definition as introduced by Hobbes, is for Leibniz a real definition in an exemplary way because it displays the demanded possibility of its subject. Leibniz does not even mention any other type of real definitions. He then somewhat laconically concludes: “Hence it is useful to have definitions involving the generation of a thing, or if this is impossible, at least its constitution, that is a method by which the thing appears to be producible or at least possible” (L 230–31; A 6, 4, N. 129, 541). Just as Hobbes and Spinoza did, Leibniz here extends the scope of causal definitions by way of any construction of a thing even if its actual cause might have been another one. Any construction of a thing that can generate it provides a clear guarantee of its possibility.

In the same work, Leibniz provides an explicit statement on the relation between an adequate idea and a genetic or causal definition: such genetic definitions are adequate ideas because they immediately display the possibility of the defined thing, that is, without an experiment or test or observation, as well as without the need to show the possibility of something else in advance. Such an adequate idea is given whenever the thing can be analyzed into its simple primitive concepts, which is precisely the case in geometrical causal definitions. “Obviously, we cannot build a secure demonstration on any concept unless we know that this concept is possible.… This is an a priori reason why possibility is a requisite in a real definition” (L 231; A 6, 4, N. 129, 542).

It is somewhat ironic that Leibniz uses this opportunity to criticize Thomas Hobbes for having claimed (as indeed he did) that all definitions are arbitrary and nominal. Leibniz knew full well that Hobbes also provided the other type of causal definitions, as mentioned earlier, which are not arbitrary. Leibniz continues that we “cannot combine notions arbitrarily, but the concepts we form out of them must be possible … Furthermore, although names are arbitrary, once they are adopted, their consequences are necessary, and certain truths arise which are real even though they depend on characters which have been imposed” (ibid.).

Leibniz emphasizes the necessity of the consequences as they follow from adequate ideas—that is, from causal definitions. These adequate ideas or genetic definitions are further praised for their special capacity: “From such ideas or definitions, then, there can be demonstrated all truths with the exception of identical propositions, which by their very nature are evidently indemonstrable and can truly be called axioms” (ibid.). True axioms are exclusively identical propositions. How close Leibniz is to Hobbes here can be seen in the following sentence in which he makes the unusual claim that even common axioms can actually be demonstrated—which Leibniz had in fact learned from Hobbes very early in his philosophical career (Goldenbaum 2008, 53–94). His critical statement against arbitrary definitions may thus have been directed at those critics of the geometrical method who questioned it for its uncertain starting point—the axioms.

As is well known, Leibniz makes another bold claim, not so horrifying to mathematicians but to theologians. He says that a reason can be given for each truth “for the connection of the predicate with the subject is either evident in itself as in identities, or can be explained by an analysis of the terms. This is the only, and the highest, criterion of truth in abstract things, that is, things which do not depend on experience—that it must either be an identity or be reducible to identities” (L 232; A 6, 4, N. 129, 543).32 From here, Leibniz states, the elements of eternal truths can be deduced and a method provided for everything if they are only cognized as demonstratively as in geometry. Of course, God cognizes everything in this way, that is, a priori and “sub specie aeternitatis,” because He does not need any experience. While He knows everything adequately and intuitively, we can grasp hardly anything in this way and have to rely on experience. Notwithstanding, Leibniz then recommends the development of empirical sciences that combine a priori knowledge with experiment in mixed sciences, which are supposed to enrich human knowledge. There is no question that Leibniz walks thereby precisely in the paths of Galileo, Hobbes, and Spinoza, being much less cautious in terms of theology than Descartes.

AVOIDING THE NECESSITARIANISM OF THE GEOMETRICAL METHOD

So, what is so problematic about the geometrical method? According to this new geometrical method, which epistemologically goes far beyond Euclid, we, as humans, can have a priori knowledge, considered by rationalists (with the exception of Hobbes) to be divine knowledge, although of a very small number of things. Because we can deduce every property from genetic definitions, the degree of certainty of our knowledge of these things will be no less than that of God’s knowledge, although He, of course, knows everything intuitively while we know it for the most part by the hard work of demonstrations. What is more challenging even, is that the converse is true as well: according to the new geometrical method, God’s capacity of knowing things functions in the same way as that of our knowing. It is because He constructed/created all things in the universe that He knows them all a priori. It is only our finiteness and our limited capacity for intuition that hinders us from knowing everything a priori like the master geometrician God. As a result, the difference in knowledge between us and God would not be ontological but merely a difference in degree. That is precisely what Galileo had claimed (who is often quoted but not as often fully understood); namely that the book of nature is written in mathematical signs (1960, 183–84). Taking the knowledge of everything through causal definitions (that is, adequate ideas) into the scope of divine knowledge opens an avenue for the endless extension of human knowledge far beyond geometry.

But this avenue, potentially leading to necessary knowledge about everything, seemed to lead into strict determinism, thus threatening free will. This can be seen in the cases of Hobbes and Spinoza, who both were straight determinists. In contrast, it was precisely the recognition of this threat of determinism that led Henry More to his rejection of Descartes and even more of Cartesianism.33 Leibniz, embracing the geometrical method, was fully aware of his dangerous intellectual neighbors (heretics and determinists), and worked hard to secure his metaphysics against strict determinism in order to distinguish his metaphysical and epistemological project from theirs. He had been working on this since he first studied Hobbes and Spinoza in Mainz between 1670 and 1672. The result is his well-known distinction of necessitating versus inclining at the end of the heading of paragraph 13 of the Discourse on Metaphysics. But, notwithstanding his obvious rejection of Hobbes’s and Spinoza’s strict determinism, Leibniz clearly shares the new geometrical method, as a philosophical method, with the infamous philosophers. Moreover, it is this new method based on the genetic or causal definition that provides the basis of Leibniz’s logic of containment (Di Bella 2005, 80–95).

While only God can have a priori knowledge of the complete notions of individuals, we can at least have a priori knowledge of abstracta, although we have to rely on empirical knowledge when it comes to individuals (L 331–38; A II, 2, N. 14). This distinction, closely related to the distinction between necessary and contingent truths, gave Leibniz sufficient confidence to present at least the headings of his Discourse on Metaphysics to Arnauld in 1686, with the long section 13 being especially provocative in respect to free will. By this time, Leibniz had already worked out his new metaphysics (based on the problematic new geometrical method), which would make modern science compatible with Christian dogmatics and free will with (softened) determinism.

Finally, in spite of Leibniz’s strong emphasis on the different ontological status of specific/abstract truths and contingent truths (he held since the Confession of the Philosopher) and then on the logical distinction of concrete and abstract things (since 1676) both aiming to secure contingency and to block strict determinism, he always maintains the containment theory. But this view (that the predicate of a true proposition must be included in its subject) clearly retains a general similarity between the two kinds of concepts because both—specific (or full) concepts of abstract things as much as complete concepts of individuals—must include all their predicates and can be known a priori by Him who generated them. This view is the core of the geometrical method! It was this theory that would lead to paragraph 13 of the Discourse of Metaphysics, according to which the complete concept of any individual was known by God and would include every single event that would ever happen to us.

At first glance, the mere claim that we cannot know individual things by a priori knowledge as God does, but only by observation and empirical research, does not sound at all new or promising and seems simply to confirm God’s omniscience and the limits of our reason. Leibniz’s conception is more subtle, which can already be felt by the vehemence of the theologians’ protestations against him.34 According to Leibniz, even if human beings cannot know individuals a priori but only through empirical study or by history, God does know the concepts of individual substances a priori. Moreover, God chose them as belonging to the best of all possible series of things when He created this world. Because of that choice, led by God’s intellect, there cannot be any contradiction among the things of one series or one world. What is crucial here is that Leibniz’s approach to contingent things assures us—from the very beginning—of the coherence of all phenomena of this world that will ever occur to us in our experience. This is so even if we do not yet see it. Because there is nothing arbitrary in God’s creation—nihil sine ratione—we can take for granted that there is a universal coherence of the world in spite of our own limited approach. It is this view that deviates from Luther and the Protestant way of thinking, wherein which such an intelligibility of the world to humans is bluntly denied. According to that view, human reason has been corrupt since the fall and thus must fail to understand. Moreover, we cannot even know whether God would have wanted to create a coherent world. God is hidden from us and we can know about Him only through faith and revelation.

Leibniz is often said to be an optimist. The true optimism that can indeed be ascribed to the rationalist philosopher lies less in his belief that this is the best of all possible worlds than in the comprehensibility of the world based on the comprehensibility of God, thanks to the new geometrical method, based on genetic definitions. Moreover, this method not only enables us to have a priori knowledge in mathematics and other fields of merely conceptual knowledge, but provides us with a new approach to empirical research to obtain contingent truths. For Leibniz, learning the many predicates of individuals through experience does not mean simply gathering and collecting data, and watching out for common patterns from which to abstract rules. Rather, our gathered data are supposed to fit into a larger theoretical framework, known by God and—partially—by us.

This framework includes of course those full specific notions of abstract things, which we as humans are able to know a priori because their number of predicates is finite. Because these eternal abstract truths can never contradict any predicate of a complete notion, they can provide a strong framework for our empirical work, which is available to our finite knowledge. When we come to learn about new facts by experience and by history, we can expect these single historical facts to fit into the theoretical framework like the pieces of an unfinished puzzle, and build a more complete notion of an individual and its action.

Of course, this infinite process of learning can never be conclusive because it is infinite. Nevertheless, our expectation (based on the conviction of a theoretical framework that is known by God a priori and thus exists) together with the available specific notions of abstract things we have at hand a priori, provides powerful tools. It is as if we had an unfinished map, a compass, and a watch, which, with our general framework of terrestrial geography, can guide an expedition into an unknown area. Such equipment can help us to recognize coherence and causal interconnectedness in the otherwise confusingly rich abundance of single facts of empirically obtained knowledge. Therefore, Leibniz’s (Spinoza’s and Hobbes’s) approach to empirical research is completely different from any empiricist approach to nature or history, the latter being a mere collecting of facts while looking for patterns or similarities to abstract from them, and thus to find rules.

While we distinguish between natural science as hardcore science (such as physics, chemistry, biology or, increasingly, medicine) on the one hand and humanities and social sciences on the other, Leibniz (as well as Hobbes and Spinoza) instead distinguishes demonstrative knowledge (using specific notions and dealing with abstract things such as geometrical figures) from the empirical sciences (relying on empirical knowledge in addition to a priori knowledge). Thus for Leibniz, human history and the humanities are not really different from natural sciences in their searching for factual and contingent truths connected through a theoretical framework of a priori eternal truths available to us through the geometrical method. He was confident that empirical knowledge can be turned into science through a theoretical framework of a priori knowledge. This stands in sharp contrast to Locke. Leibniz is following the path of Hobbes and Spinoza, and he will be in turn followed by Wolff and Tschirnhaus.

CONCLUSION

The trouble with the geometrical method in the seventeenth and eighteenth centuries was neither its ponderous way of thinking nor its lack of success. Rather it was the turmoil about human haughtiness and the threat that its determinism would destroy free will, the arbitrary choice of the will of God as well as that of human beings. The correspondence of Leibniz and Clarke exemplifies the different approaches to God’s free will. According to Leibniz, nothing can happen without a sufficient reason, which proves the existence of a God, who in His perfection could not have chosen an arbitrarily functioning world. Clarke (and Newton), on the other hand, count any act of will on God’s part as a sufficient reason (Leibniz-Clarke, 11; 2nd Reply, #1).

Two things caused deep anxiety and anger regarding this method: (1) the attempt to extend the geometrical method to nature, to humans, and to society (taking mathematization of nature for granted), thus providing human beings with a God-like a priori knowledge beyond mathematics; and (2) the threat of determinism. These threats forced theologians and Christian philosophers to reject rationalism and the geometrical method altogether. In sharp contrast to rationalism, Locke would even deny the possibility of natural science because we could not have any real definitions beyond mathematics and morals: “This way of getting and improving our Knowledge in Substances only by Experience and History, which is all the weakness of our Faculties in this State of Mediocrity, which we are in this World, can attain to, makes me suspect, that natural Philosophy is not capable of being made a Science. We are able, I imagine, to reach very little general Knowledge concerning the Species of Bodies, and their several Properties” (Locke 1975, 645; Essay 4, 12, 10). Kant would declare that there would “never be a Newton for a blade of grass” (Kant 2000, 268–71), pointing us instead to design theory in biology admitting causal explanations alone for mathematics and mechanics, or applied mathematics.

Thus, the opposition between the two philosophical camps of rationalism and empiricism was not the result of different approaches to experience as is often claimed. Rather, it was their different and opposing stances toward the geometrical method and the mathematization of nature. This method was in no way external to rationalist philosophy. As much as rationalist philosophers differ in their philosophical systems, they all agree that human beings can arrive at a priori knowledge (through deducing from definitions), independent of experience, and that this knowledge is somehow “divine,” that is, as certain as God’s knowledge. In contrast, empiricists and theologians are eager to deny such a possibility, and therefore must rely exclusively on knowledge by experience. To be sure, empiricists do not trust experience any more than rationalists do. Rather they deny that we are capable of any better knowledge (except within mathematics). Thus it is the different approach to the new geometrical method that provides the explanation for the two schools of early modern philosophy. While rationalists see the mathematization of nature and the geometrical method as avenues to comprehend God’s creation, sharing a priori knowledge with Him, the empiricists ally with the Christian belief (emphasized more strongly by Protestants) that human reason is corrupted due to the fall and that God as well as the essences of the created things are hidden from us. The geometrical method of the early modern period was much more than a way of demonstration. It was a new epistemological approach to true knowledge of the external world, based on the mathematization of nature, completely different from any traditional, empirical approach of natural philosophy.

ABBREVIATIONS

NOTES

1. The literature about the geometrical method in early modern philosophy is to a large extent focused on Spinoza’s Ethics. See though Cassirer 1974, 1:136–44, 512–17, and 2:48–61, 86–102; Schüling 1969 (who confuses geometrical method with axiomatic method); see also Hecht 1991. On Spinoza’s use of the geometrical method, see Hubbeling 1964, 1977; Curley 1986a; and De Dijn 1986. See also Curley 1986b; Klever 1986; Matheron, 1986; and Goldenbaum 1991.

2. Due to the common confusion of axiomatic and geometrical method, Wolters even sees Spinoza’s Ethica more geometrico demonstrata as exemplary for the degeneration of the axiomatic method into a mere external tool of presentation. See Wolters 1980, 7.

3. This was the view of Hegel and the German romantics which has been canonized in the influential German history of philosophy shaped by Hegel’s view, as for example in Windelband: “The deep motion of a god-filled mind is expressed in the driest form, and the subtle religiosity appears in the stiff armor of fixed chains of conclusions” (Windelband 1919, 212; my translation).

4. Breger calls the concept of motion an essential driving engine for the conceptual transition of mathematics of the seventeenth century, especially in the development of the concept of function. He continues: “The concept of motion has not only paved the way to the problem of rectification, to the introduction of the transcendent, and as a tool to investigate limit processes; it also contributed to legitimizing the continuum (and thereby eventually infinitesimal methods). The concept of motion makes it implausible that the continuum could have gaps. The mechanical thinking makes the geometrical lines appear as homogenous and all points on them as equally justified: the limitation to points and lines which can be constructed in this or that way, no longer appears as a necessary condition of exactness but as an unnatural limitation, which was to be overcome through a new boundary line between mathematics and mechanics” (Breger 1991, 45; my translation).

5. “To clarify this fact one has to reflect on the modern form of geometry Spinoza had in mind. In fact, it is not the Euclidean but Cartesian geometry that is the systematic model for him. In analytic geometry, the number refers to space, i.e., a mere mode of ‘thinking’ refers to a mode of ‘extension’ in such a way that a gapless, one-to-one correspondence happens between both. Every dependence between figures in space is mirrored in a dependence between quantities in numbers: thus here one and the same connection is expressed in two different forms” (Cassirer 1974, 117–18; my translation); “En fait, l’idée du cercle est une image, resçue par’ l’esprit, une peinture faite à l’imitation d’un modèle externe; l’idée cartésienne du cercle est un concept né de l’activité proprement intellectuelle, de la force native de l’esprit. Le cercle et son idée appartiennent à deux orders different” (Brunschvicg 1904, 771).

6. “But because the almighty God gave the ability to us human beings to perfectly conceive the numbers and quantities, and did not keep anything for Himself, we easily fall into the awkward thought that we could also be the master of all the other objects of our knowledge (cognoscibilia), and could have the sufficient reasons under our control” (Löscher 1735, 119; my translation).

7. “I only say this […] that the author deduces the stubbornness [of Pharaoh in Exodus 7, 13 and following] from the nexus or the fatal connection of all things, and in this way ascribes it to God according to his preestablished harmony. This nexus is the soul of the whole system of the mechanical philosophy” (Lange 1735, 25; my translation). This is directed against the author of the Wertheim Bible, Johann Lorenz Schmidt, who had produced a Wolffian translation of the Pentateuch; cf. Goldenbaum 2004, 236.

8. “Yes, one should accept, so to speak, only genetic demonstrations, or those that are taken from the generation of the subject, so that one will know in advance perfectly, how the subject came about.… These demonstrations are considered the only ones that provide science, a true knowledge: from this it follows that all other knowledge, proved in other ways, is opinion only, and cannot be trusted” (Löscher 1735, 126; my translation).

9. “Accordingly, we would have only nominal definitions of God and many other objects from where one could not even see whether the thing is possible [rem esse possibilem]” (Löscher 1735, 129; my translation).

10. “It is an obvious pedantry, if one plants oneself with one’s mathematical method in other disciplines so broadly; the most rude thing, though, is doing such a thing with the Holy Scriptures and (N.B.) in theology” (Lange 1735, 2, §4; my translation).

11. “One takes such a reason not only to be a reason, or puts it as such in the intellect. It is supposed to be in reality as well, and not in any different way. ‘Sufficient,’ in this philosophy, does not mean something what we can be content with, knowing it according to its constitution. Nay, it means something so strong, perfect, and adequate, that it is sufficient everywhere, and nothing more can be asked without lacking reason” (Löscher 1735, 119; my translation).

12. “6. Asserirsi e dichiararsi male qualche uguaglianza, nel comprendere le cose geometriche, tra l’intelletto umano e divino” (Dok. 20; Car. 387r–393r).

13. Galileo makes Salvati refer to “our Academician,” obviously Galileo himself, “who had thought much upon this subject and according to his custom had demonstrated everything by geometrical methods so that one might fairly call this a new science. For, although some of his conclusions had been reached by others … they had not been proven in a rigid manner from fundamental principles. Now, since I wish to convince you by demonstrative reasoning rather than to persuade you by mere probabilities, I shall suppose that you are familiar with present-day mechanics so far as is needed in our discussion” (Galilei 1954, 6).

14. Thus it became the strategy of Rüdiger, Hoffmann, Crusius, Löscher, and Lange to emphasize the fundamental difference between mathematical knowledge and scientific knowledge of natural things whereby mechanical theory counted as applied mathematics. Whereas mathematics dealt with figures and numbers—produced by humans and thus arbitrarily—natural science, as well as metaphysics and theology, dealt with God’s creation and thus with natural things (see Löscher 1735–42, 128–29; 1742, 78). Therefore, only mathematical concepts could be known by us in their very essence, whereas the essences of God’s creatures remained hidden to us. We could know them only by observation, experience (equated with sense perception), induction, and abstraction. This approach well explains the enthusiasm of German Pietism for Locke.

15. The manuscripts of Leibniz’s Elementa Iuris naturalis can be seen as an exercise in demonstrating by chains of definitions (A VI, 1, N. 12). See also Leibniz’s letters to Chapelain from the first half of 1670, in A VI, 1, N. 24.

16. There are various attempts to avoid the rationalist geometrical method by modernizing scholastic philosophy through empirical research, adopting elements of modern science while attacking the geometrical method, that is, the central role of causal definitions, if used beyond geometry. Thus the L’Essai de logique, published by Abbé Edme Mariotte, a gifted experimenter (coauthored by Roberval) appears as a turn against Galileo’s and Descartes’ mathematization of nature. Our knowledge of nature is restricted to observation and experiment (cf. Roux 2011, 63–67), a clearly empiricist move against disciples of Descartes. Mariotte considers even the principle of inertia as the result of experience (ibid., 104). On Roberval and Mersenne, see also Fouke (2003, 75–76).

17. “What I want to emphasize about this passage is that in it Spinoza shows himself to be willing, when one of his fundamental assumptions is questioned, to provide further argument for this assumption. He does not regard his axioms as argument-stoppers, principles so fundamental that they neither require nor can be given any further argument. Instead, he offers to demonstrate his axioms by appealing to his definitions. It is interesting that in the final version of the Ethics all four of these axioms are removed from the list of fundamental assumptions. Three become propositions. One becomes a step in a demonstration” (Curley 1986b, 157); see also Klever (1986).

18. In full agreement with Lutheran theologians and with Crusius, Kant argues that philosophy cannot begin with definitions because its objects do not depend on human minds as the objects of mathematics do [CrR B740–763]. He has to ignore however, to make this argument, that Wolff (and other rationalists) are fully aware of this problem and indeed make the production of good definitions of real/natural things a task that has to precede any demonstration. In case no sufficiently clear and distinct definition can be found, a nominal definition can serve as a placeholder for the time being from which hypothetical knowledge can be deduced as long as no contradictions emerge. This is seen, to some extent, by Engfer (1982, 56).

19. Thus, the young Kant sharply distinguishes between mathematical bodies and natural bodies whereby the former do not have any internal force while the latter in fact do own such a force: The latter “has a power in itself, through itself to enlarge the force which was awakened in it by an external cause of its motion, thus that it can include grades of force which did not originate from external cause of motion and which are larger than it. Therefore they cannot be measured by the same measure as the Cartesian [mechanical] force and have another estimation” (KAA 1 140 §115).

20. “On ne reconnaît en géométrie que les seules definitions que les logiciens appellant défintions de nom, c’est-à-dire que les seules impositions de nom aux choses qu’on a clairement designées en termes parfaitement connus; et je ne parle que de celles-là seulement.… D’où il paraît que les definitions sont très libres, et qu’elles ne sont jamais sujettes à être contredites; car il n’y a rien de plus permis que de donner à une chose qu’on a clairement désignée un nom tel qu’on voudra. Il faut seulement prendre garde qu’on n’abuse de la liberté qu’on a d’imposer des noms, en donnant le meme à deux choses différentes” (Pascal 2000, 156).

21. “Nous connaissons la vérité non seulement par la raison mais encore par le cœur. C’est de cette dernière sorte que nous connaissons les premiers principes et c’est en vain que le raisonnement, qui n’y a point de part, essaie de les combattre” (Pascal 2000, 573).

22. The “mechanical” definition by mechanical motion to produce a geometrical object occurred accidentally in ancient mathematics, not in any systematic, conscious way though. See Breger 1991, and on Hobbes and Roberval see Jesseph 1999, 117–25; 1996, 86–92.

23. “Hobbes does not think anymore of motion as an inner quality and constitution of bodies but as a mere mathematical relation, which we can construe on our own and therefore conceive. With this one step, the transition from Bacon to Galileo is accomplished. The analysis of natural objects does not end in abstract ‘entities’ but in laws of the mechanism, being nothing else but the concrete expressions of the laws of geometry” (Cassirer 1974, 2:47–48).

24. While Leibniz hardly used this method explicitly (cf. though his treatise in favor of the election of the Polish king, in Leibniz [1924–ongoing, 3–98]; A IV, 1, N. 1), he completely agreed with Hobbes about demonstrations as mere chains of definitions (Leibniz A VI, 1, N. 12; see also Leibniz A II, 1, N. 24, 153).

25. “In his quatuor partibus continetur quicquid in philosophia naturali, demonstratio proprie dicta explicari potest. Nam si phaenomenωn naturalium speciatim causa reddenda sit, puta quales sint motus, et virtutes corporum cœlestium, et partium ipsorum, ea ratio ex dictis scientiae partibus petendea est, aut omnino ratio non erit, sed conjectura incerta” (OL 1 62–65; De corpore i, 6, §6). “Scientia intelligitur de theorematum, id est, de veritate consequentiarum. Quando vero de veritate facti agitur, non proprie scientia, sed simpliciter cognitio dicitur. Itaque scientiae a quidem, qua scimus propositum aliquod theorema esse verum, est cognitio a causis, sive a generatione subjecti per rectam ratiocinnationem derivate” (OL, 2:92; De homine ii, 10, §4).

26. “And as the art of well building is derived from principles of reason, observed by industrious men that had long studied the nature of materials and the divers effects of figure and proportion, long after mankind began (though poorly) to build, so, long time after men have begun to constitute commonwealths, imperfect and apt to relapse into disorder, there may principles of reason be found out by industrious meditation, to make their constitution … everlasting. And such are those which I have in this discourse set forth” (Hobbes 1994, 220; Leviathan xxx, 5).

27. I take it to be an understatement even when Curley states: “I am not persuaded that Spinoza was such a radical anti-empiricist” (1986b, 156). In addition to Spinoza’s own desire to develop a theory of experimentation (TIE 102–3; C 42), we also have evidence from Tschirnhaus via Wolff that Spinoza experimented himself (cf. Corr 1972, 323–34).

28. “Pour les veritez eternelles, je dis derechef que sunt tantum veræ aut possibiles, quia Deus illas veras aut possibiles cognoscit, non autem contra versa à Deo cognosci quasi independenter ab illo sint verae” (AT, 1: 145, 149–50; Descartes to Mersenne, April 15, 1630 and May 6, 1630).

29. “Seule la géométrie cartésienne permet de rapporter la vérité à l’autonomie de l’intelligence; les propriétés d’une courbe se déduisent en effet de la définition analytique de cette courbe, c’est-à-dire d’une équation abstraite, sans recours à la considération directe de la figure. Seule elle permet d’interpréter la notion spinoziste de la convenance. La convenance n’implique plus l’antériorité de l’objet par rapport au sujet, mais la correspondence du sujet qui comprend et de l’objet qui est endendu, le parallélisme de deux orders d’existence qui ne suffisent à eux-mêmes, qui n’interfèrent jamais” (Brunschvicg 1904, 772). Therefore, adequatio in Spinoza is understood completely differently from scholastic tradition, well known by Spinoza according to Brunschvicg.

30. “The Divine intellect, by a simple apprehension of the circle’s essence, knows without time-consuming reasoning all the infinity of its properties. Next, all these properties are in effect virtually included in the definitions of all things; and ultimately, through being infinite, are perhaps but one in their essence and in the Divine mind. Nor is all the above entirely unknown to the human mind either, but it is clouded with deep and thick mists, which become partly dispersed and clarified when we master some conclusions and get them so firmly established and so readily in our possession that we can run over them very rapidly” (Galilei 1967, 103–4; emphasis added). This is very similar to Descartes’ understanding of intuition as not exclusively instantaneous but also as a quick running through: “necesse est illas iteratâ cogitatione percurrere, donec à primâ ad vltimam tam celeriter transierim, vt fere nullas memoriae partes relinquendo rem totam simul videar intueri” (AT 10 409 [Reg. XI]).

31. I have discussed this momentous innovation of Leibniz elsewhere; see Goldenbaum 2011a.

32. Loemker translates “justify” instead of “giving a reason,” which sounds to me more like Hume than Leibniz.

33. “Sed si ullubi magnopere culpandus sit nobilissimus Philosophus, ob illud potissimùm eum reprehendum censeo, quòd Mathematico suo Genio ac Mechanico in Phenomenis Naturae explicandis nimium quantum indulferit. Eam tamen interim agnosco summorum Ingeniorum felicitatem, ut vel vitia eorum & errores aliquam virtutis speciem habeant atque fructum. Et profectò mihi planè incredibile videtur, nisi ingentem illam spem concepisset demonstrandi Omnia ferè Mundi Phaenomena ex necessariis Mechanicae legibus, eum unquam tot tantàque tentare voluisse, aut tentata potuisse perficere” (More 1711, 58).

34. The following argument of a student of the influential Lutheran (Pietist) theologian Budde addresses only one although central point of criticism: “Nam tunc Deus mundum non eligit, quia optimus est, sed optimus est, quia eligit.… Hinc quidquid Deus elegit … non est optimum moraliter per se, sed ob Dei electionem” (Budde 1712, 72, §5).

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