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date: 20 October 2019

Spinoza and the Philosophy of Science: Mathematics, Motion, and Being

Abstract and Keywords

This chapter argues that the standard conception of Spinoza as a fellow-traveling mechanical philosopher and proto-scientific naturalist is misleading.1 It argues, first, that Spinoza’s account of the proper method for the study of nature presented in the Theological-Political Treatise points away from the one commonly associated with the mechanical philosophy. Moreover, throughout his works Spinoza’s views on the very possibility of knowledge of nature are decidedly skeptical. Third, in the seventeenth-century debates over proper methods in the sciences, Spinoza sided with those who criticized the aspirations of the physico-mathematicians like Galileo, Huygens, Wallis, and Wren who thought the application of mathematics to nature was the way to make progress. In particular, he offers grounds for doubting their confidence in the significance of measurement as well as their piecemeal methodology. Along the way, this chapter offers a new interpretation of common notions in the context of treating Spinoza’s account of motion.

Keywords: Motion, common notions, philosophy of science, philosophy of mathematics, essence

“Being finite is really, in part, a negation.”



This chapter argues that the standard conception of Spinoza as a fellow-traveling mechanical philosopher and proto-scientific naturalist is misleading.1 It argues, first, that Spinoza’s account of the proper method for the study of nature presented in the Theological-Political Treatise points away from the one commonly associated with the mechanical philosophy. Moreover, throughout his works Spinoza’s views on the very possibility of knowledge of nature are decidedly skeptical (as specified herein). Third, (p. 156) in the seventeenth-century debates over proper methods in the sciences, Spinoza sided with those who criticized the aspirations of the physico-mathematicians such as Galileo, Huygens, Wallis, and Wren who thought the application of mathematics to nature was the way to make progress. In particular, he offers grounds for doubting their confidence in the significance of measurement as well as their piecemeal methodology (see section 2). Along the way, this chapter offers a new interpretation of common notions in the context of treating Spinoza’s account of motion (see section 3).

Scholarship on Spinoza routinely portrays him as a second-generation, fellow traveler of the so-called mechanical philosophy, that is, the intellectual movement that sees the world as a machine and aims to explain natural phenomena with reference to the size, shape, and motion of bodies.2 Besides offering a very intelligent introduction to it in DPP, Spinoza was familiar with the aspirations of that program in the Royal Society (see Ep. 3). Descartes and Boyle are, despite their disagreements, often taken to be paradigmatic mechanical philosophers.3 Spinoza also thought of Bacon as one the project’s founders (Ep. 6). Within the mechanical philosophy, mathematical laws of motion and the rules of collision are the foundational explanatory principles. During Spinoza’s lifetime, in 1669, Huygens, Wallis, and Wren rejected Descartes’s foundational approach, and, despite some subtle differences in their metaphysical conceptions of space and motion, independently established a consensus concerning the proper mathematical formulation of the rules of collision; it was claimed that these had sufficient empirical confirmation. In the Principia (a decade after Spinoza’s death) Newton hailed their breakthrough. Spinoza disagreed with at least one of Descartes’s collision rules (the sixth), and he seems to have been unimpressed by Huygens’s arguments and the empirical claims on its behalf (see Ep. 30A). This should alert us to realizing that Spinoza’s relationship to the mechanical philosophy is not straightforward.

Moreover, in recent scholarship Spinoza is also nearly always treated as a kind of scientific naturalist. Spinoza’s immersion and evident interest in the world of natural philosophy is illustrated by his correspondence with Henry Oldenburg, the (p. 157) secretary of the Royal Society, and (indirectly through him) Robert Boyle; by his proximity to and regular contact with the Huygens brothers; by the known reports of his experiments; by his adoption of terminology inherited from Cartesian mechanics; by his lens-crafting; by his knowledge of optics (and with it state-of-the-art knowledge of microscopy and telescopes);4 by his debunking of reported miracles as signs of epistemic ignorance (Chapter 6 of TTP and also Ep. 73, 75); by his attack on superstition and final causes; and by his library full of up-to-date works on natural philosophy. All these tend to suggest that Spinoza should be understood in terms of an arc that originates in Galileo, Bacon, Descartes, and Hobbes and that leads if not toward Newton or modern quantum field theory5 then at least toward Leibniz’s dynamics.6 This reading fits seamlessly into the now discredited attribution to Spinoza of two short pieces on probability and the rainbow—two topics central to the new focus on the mathematization of nature and society.7 More recently, the standard reading has received indirect support and reinforcement from the tendency to read Spinoza as source of (radical) Enlightenment thought, which is taken to be “pro-science.”8

One problem the standard interpretation faces is Spinoza’s near-complete absence in works on the history of science. Even the great Dijksterhuis, who was not shy about noting the Dutch contribution toward the mechanization of the universe, fails to mention Spinoza. This is by no means a fatal objection to the standard reading. After all, it requires not that Spinoza made contributions to the new science but that he was a fellow traveler in the program. In response, the defenders of the standard reading can point to Spinoza’s authorship of what we may call a leading textbook introduction to Cartesian physics (DPP). Textbook writers need not be on the cutting edge of science. Moreover, DPP is no slavish summary of Descartes, but it offers genuine innovations (p. 158) on Descartes’s Principles.9 Moreover, there is evidence that Spinoza was collaborating with Johannes Hudde, then one of Europe’s foremost mathematicians on building a very powerful telescope (see the closing paragraph of Ep. 36).10

Nevertheless, the standard reading has had to ignore some inconvenient evidence about the eighteenth-century reception of Spinoza; Newtonians were very eager to distance Newton from Spinoza and provided some of the most informed and detailed criticism of Spinoza’s metaphysics and physics.11 While the motives of these critics may have been religious or social (which explains some of the vehemence of their attacks on Spinoza) and their criticism may have been in some respects anachronistic (after all, Spinoza could not have anticipated Newton), the existence of the Newtonian rejection of Spinoza alerts us to the fact that at least one group of informed natural philosophers did not consider Spinoza as a fellow traveler at all. Of course, Newtonians objected to Cartesian physics more generally, so this criticism is in some respects to be expected. However as I argue here, some of their criticism alerts us to the shortcomings in Spinoza’s conception of motion in particular.

1.  Knowledge of Nature

In this section I analyze Spinoza’s proposed, sophisticated method for empirical inquiry into nature. In particular, I characterize Spinoza’s rather pessimistic stance on our ability to have knowledge of the physical world. In doing so I analyse what Spinoza means by definition and how it relates to empirical inquiry.

1A.  Method: Empirical Inquiry into Nature

In a letter to Blyenbergh, Spinoza wrote, “Ethics, … as everyone knows, ought to be based on metaphysics and physics” (Ep. 38). Yet in the Ethics Spinoza is surprisingly terse about the nature of physics and its relationship to metaphysics and ethics. We learn little explicitly about their inner relationship and their methodologies. However, in Chapter 7 of the Theological-Political Treatise, Spinoza elaborates on scientific method, so I turn there first.

In the context of explaining his method of interpreting Scripture, Spinoza says:

[It] does not differ at all from the method of interpreting nature, but agrees with it completely. For just as the method of interpreting nature consists above all in putting together a history of nature, from which, as from certain data, we infer the definitions (p. 159) of natural things, so also to interpret Scripture it is necessary to prepare a straightforward history of Scripture and to infer the mind of the authors of Scripture from it, by legitimate reasonings, as from certain data and principles. (TTP 7/G 3:98)12

This passage has attracted a lot of attention from people who wish to understand Spinoza’s controversial reading of the Bible.13 But here I focus on what it implies about what Spinoza thinks about the study of nature.

At first, Spinoza suggests that the study of nature consists of two inductive steps. First we create a history, and second we infer from it the definitions of things. I discuss the meaning of these crucial terms in light of Spinoza’s natural philosophy and metaphysics in turn. From what Spinoza says a few paragraphs down (“collect the sayings of each book and organize them under main headings so that we can readily find all those concerning the same subject”) about how to approach Scripture we can infer that in the context of inquiry, by history Spinoza means creating lists or tables of natural events ordered by topic. As Alan Gabbey points out, this sounds like a step in the method of natural history Bacon promotes.14 From Ep. 2, we can infer that Spinoza had read Bacon’s New Organon. If we take the strict analogy between the study of nature and the interpretation of scripture seriously, then Spinoza also means to imply that we carefully note the circumstances in which events are recorded and transmitted to use (cf. TTP 7/G 3:101).

Now it is easy to ridicule this extreme inductivism, but Spinoza offers a number of constraints on it. For example, “in examining natural things we strive, before all else, to investigate the things which are most universal and common to the whole of nature—viz., motion and rest, and their laws and rules, which nature always observes and through which it continuously acts and from these we proceed gradually to other less universal things” (TTP 7/G 3:102). Rather than making lists of everything, the inquiry of nature should focus on the study of motion and rest and their laws and rules because it is most universal and common. (In 3B I explore such common notions.) Two important points follow from this: first, the study of motion and of rest is foundational; second, if one were to know the laws of motion and rest one could use these to constrain subsequent research. These points make Spinoza appear to be a mechanical philosopher.

Moreover, to readers accustomed to thinking of Spinoza as offering a great deductive system, it must be tempting to go a step further and suggest, third, that Spinoza proposes we deduce all other phenomena from the laws of motion; in TTP he does not advocate this position unambiguously.15 Spinoza’s Political Treatise suggests that there is indeed a deductive step after we have relied on experience (induction) to reach proper understanding of things (i.e., definitions; see TP 1.4; TP 2.1; TP 3.1). But there is no (p. 160) evidence that this deduction proceeds from the laws of motion or collision. In fact, in TdIE Spinoza insists that “from universal axioms alone the intellect cannot descend to singulars [singularia], since axioms extend to infinity, and do not determine the intellect to the contemplation of one singular thing rather than another” (TdIE §93). That is, the inductive and deductive steps are connected by and come together in “true and legitimate” definitions of created beings—not the laws of motion. In fact, TdIE is quite explicit that “we ought to seek knowledge of particulars as much as possible” (TdIE §98; Unde cognitio particularium quam maxime nobis quaerenda est.)16 Of course, TdIE appears as an incomplete work, but as I show there is little reason to think Spinoza changed his mind fundamentally on the main issues treated in this chapter.

Much ink has been spilled in relating Spinoza’s mechanical philosophy to Descartes’s program for the sciences.17 But it has been little noticed that Spinoza seems to have had no interest in articulating the laws of nature. In fact, when Spinoza deals with Descartes’s laws of nature in DPP he does not even label them laws!18 In Spinoza’s mature works there is no indication that he thinks of “laws of nature” as explanatory principles (or Cartesian “secondary causes”). If anything he seems to have been a nominalist about laws of nature (TdIE §101).19 Of course, some might see in Spinoza a nominalist of quite a general sort. However, despite Spinoza’s attacks on Platonic forms and Aristotelian universals (E2p40s1), Spinoza does believe that there are natures—for example, a Causa Sui has a nature (E1d1), and so do humans (E4p19). None of this is to deny that Spinoza often talks of the laws of nature. Yet on close inspection Spinoza uses law talk to convey the idea that nature is, first, without exception unchanging or immutable, and, second, necessary (TTP 6/G 3:86; TTP 6/G 3:83; TTP 4/G 3:58). Spinoza’s rejection of caprice in nature has mistakenly been read as a commitment to laws being foundational in one’s science.

However, as we have seen, the passage just discussed (viz. TTP 7/G 3:102) offers some evidence for the thought that in a restricted sense Spinoza is a mechanical philosopher—he, too, thinks that we should aim to understand the laws of motion and rest. In Ep. 6, he claims that they explain “nature as it is in itself” (and these laws are contrasted with ways of knowing nature derived from empirical study of nature, such as visible, invisible, warm, cold, and fluid). Moreover, in the same letter he appeals to the “proofs” supplied by “Bacon and later Descartes” in defense of the mechanical explanatory principles, that is, motion, shape, and size (to ridicule Boyle’s new experimental proofs).20 But given what he says at TdIE §93, it’s clear one cannot deduce particular facts from the (p. 161) laws of motion.21 (I return to the relationship between nature as it is in itself and empirical inquiry in a later section.)

Spinoza puts another constraint on the study of nature: “the definitions of natural things are to be inferred from the different actions of nature” (TTP 7/G 3:99). So in understanding nature we cannot rely on, say, revelation in interpreting it. In historical context this seeming throwaway line is an essential matter because it opens the door to, for example, the endorsement of Copernicanism on empirical grounds. In the previous chapter of TTP, in his treatment of the miracle of Joshua, Spinoza had already ridiculed the idea “that the sun moves, as they say, with a daily motion and that the earth is at rest” (TTP 6/G 3:92). It fits Spinoza’s more general aim to free philosophy from its role as handmaiden to theology.

For our present purposes the main significance of this remark lies elsewhere. In context Spinoza insists that definitions of natural things are arrived at only through studying how nature behaves.22 He offers his reader no Cartesian shortcuts through reason or divinely implanted innate ideas.23 Indeed, later in the book in summarizing chapter 7 Spinoza insists that “the universal history of Nature … is the foundation … of Philosophy” (TTP 15/G 3:185). That is, the study of nature is, in significant part, an empirical affair in Spinoza.24 Spinoza’s commitment to empirical inquiry is illustrated by Ep. 41 to Jarig Jelles, in which Spinoza describes an experiment he performed with two others to establish water pressure in a tube. It is no aberration because in Ep. 6 to Oldenburg Spinoza offers considerable experimental evidence against Boyle’s doctrines. From the letter it appears Spinoza had performed these experiments to test Boyle’s analysis.

In a letter to Simon De Vries Spinoza offers a sharp distinction between two domains of inquiry: (i) empirical inquiry is necessary when we are dealing with beings whose existence cannot be derived from their definitions (see also TP 2.1); and (ii) empirical inquiry is pointless when we are dealing with beings whose existence cannot be distinguished from their essence—in those cases existence can be derived from the given definitions.25 Spinoza then adds, crucially, that experience cannot teach us anything about the essences of things (Ep. 10). Little wonder that Spinoza’s impact on Locke during his stay in Holland is fertile inspiration for speculation!26 This raises interesting questions: for (p. 162) example, what is the exact relationship between, say, inductive inquiry into definitions of things and the presumably nonempirical study of essences of things? Moreover, what does Spinoza mean by definition and essence, and what is their relationship? In context Spinoza offers an interesting example of a nonempirical “eternal truth” — nothing can come into being from nothing—casually ruling out ex nihilo creation. In Ep. 10, Spinoza then insists that the things he calls “eternal truths” in accord with what he takes to be usual usage are not claims about the empirical world; rather “they do not have any place outside the mind.”27 So an additional question arises about the relationship between eternal truths, definitions, and essences.

Unfortunately, in TTP Spinoza is almost entirely silent about how to infer definitions from these tables that list the actions of nature. One available strategy would be to take Spinoza at his word about the strict methodological analogy between the study of nature and scripture (recall TTP 7/G 3:98) and analyze how Spinoza infers true meaning (even if “contrary to reason”) from the text of Scripture (TTP 15/G 3:185) and then apply it to the methodology presupposed in the study of nature. It is only an analogy because while the study of scripture is concerned with “the true meaning” the study of nature is concerned “with the truth of things” (TTP 7/G 3:100).

Taking the analogy between biblical study and study of nature seriously does provide some more clues to Spinoza’s views on method. In particular, in his scriptural method Spinoza distinguishes what we may call data from noise—he discards inconsistent and unclear utterances (TTP 7/G 3:100; see also, e.g., TTP 7/G 3:109). Presumably, this will permit the removal of a lot of entries from the tables that make up the history of nature and will encourage a search for standard measures. Furthermore, the context and source(s) of data—the entries in one’s history—must be as transparent as possible (cf. TTP 7/G 3:100–101; see also TTP 7/G 3:109–12). Finally, if we push the methodological analogy to its extreme, it appears, perhaps, that Spinoza also thinks that one must have confidence that one’s data set is not merely uncorrupted but also complete (TTP 7/G 3:106). All of this suggests that inferring definitions from the history of nature is a constrained activity.

Furthermore, according to another remark by Spinoza, this inferring of definitions from history can be done by individuals who possess what he calls “the natural light.” By this (then common locution) he means nothing mysterious: “the nature and power of this light consists above all in this: that by legitimate principles of inference it deduces and infers things which are obscure from things which are known, or given as known. This method of ours requires nothing else,” (TTP 7/G 3:112; this is very Cartesian—see Meditation 3/AT 7:38). Unfortunately, this is not very helpful in explaining how we move from the facts to their definitions. Before I analyze Spinoza’s view about definitions and essences, I say a bit more about Spinoza’s views about the scope and limits of empirical inquiry.

(p. 163) 1B.  The Scope and Limits of Empirical Inquiry

As we have seen, Spinoza clearly thinks there is an important role for empirical inquiry. We have also seen that Spinoza believes there is a method to empirical inquiry. In this section I analyze Spinoza’s attitude toward empirical inquiry and discuss how he understands its scope and limits.

In the Ethics Spinoza writes, “There is no vacuum in Nature” (E1p15s). Spinoza was familiar with air pump experiments by Pascal, Boyle, and even Huygens. In a much studied controversy Boyle argued that he was able to produce a vacuum in nature.28 Huygens repeated Boyle’s experiments successfully, but in contrast to Boyle Huygens introduced an invisible fluid to account for the so-called air-free spaces inside the tube. This is a solid Cartesian strategy to explain away the empirical evidence. It may have tempted Spinoza, too, because in Part II of the Ethics he seems to posit such fluids (see E2a3”), but he does not mention if they are invisible. Elsewhere, in one of Spinoza’s letters about Boyle to Oldenburg (Ep. 6) he seems to endorse the existence of such invisible fluids so that it is unnecessary (if not “absurd”) to posit a vacuum. Boyle certainly read Spinoza this way (Ep. 11). Yet as Huygens’s opponents in the French Academy, Roberval and Mariotte, remarked, this substitutes one mystery for another.29

In his denial of the vacuum Spinoza also pursues a second and complementary Cartesian strategy. Where the imagination or the senses see an empty space, reason knows better; if we attend to quantity30 “as it is in the intellect, and conceive it insofar as it is a substance … then … it will be found to be infinite, unique, and indivisible” (E1p15s).31 This distinction between intellectual and imaginative conception runs through the whole Ethics and Spinoza’s other works (see the first corollary to E2p44 or the whole of E2p45), I quote an important passage: “We conceive things as actual in two ways: either insofar as we conceive them to exist in relation to a certain time and place, or insofar as we conceive them to be contained in God and to follow from the necessity of the divine nature. But the things we conceive in this second way as true, or real, we conceive under a species of eternity, and their ideas involve the eternal and infinite essence of God (as we have shown in E2p45&s)” (E5p29s). When I discuss Spinoza’s reservations about the role of mathematics in inquiry, I return to this passage and the distinction between intellectual and imaginative (or imagistic) conception. For present purposes all that matters is that rational or intellectual conception does not rely on the senses.

Spinoza’s treatment of the vacuum teaches us that according to Spinoza we are not allowed to simply trust empirical perception; intellectual conception is different from and more reliable than sensory perception. To put this in terms of a slogan, intellectual conception is a (further) constraint on the deliverances of empirical perception. TTP appeals to “reason and experience” a few times (TTP 2/G 3:29; TTP 3/G 3:48; TTP 5/G (p. 164) 3:76; TTP 17/G 3:203; TTP 17/G 3:215; TTP 19/G 3:232; TTP 20/G 3:244; see also most of TTP 16).

In later generations Newtonians had great fun ridiculing Spinoza’s denial of the vacuum.32 At first glance one cannot claim that the denial of the vacuum is a central issue in Spinoza’s philosophy; in the Ethics he claims to discuss “the subject … elsewhere” (E1p15s), and he may have his earlier treatment at DPP2p3 in mind.33 Nevertheless, its significance resides in the fact that it is just about the only place where Spinoza’s Ethics is vulnerable to potentially straightforward empirical criticism.

In a late letter to Tschirnhaus, Spinoza admits that his “observations” on “motion” (i.e., mechanics) “are not yet written out in due order, so I will reserve them for another occasion” (Ep. 59). From the exchange we can discern that Tschirnhaus possessed a more or less complete manuscript of the Ethics.34 From these facts we can infer that the Ethics does not pertain to mechanics, even though Spinoza says that his system is based on metaphysics and physics (recall Ep. 38). One could object that in Part II of the Ethics Spinoza does inject what one may call a short treatise “concerning the nature of bodies” (E2p13s), also known as the “physical interlude.”35 Moreover, shortly thereafter he insists that “all those postulates which [he has] assumed contain hardly anything which is not established by experience which we cannot doubt” (E2p17s). Spinoza’s unfinished Political Treatise appeals to authority of experience throughout the opening pages (see also TTP 20/G 3:246).

Later in the chapter I offer an alternative interpretation of the physical interlude; here I focus on the status of experience in Spinoza. One should not be blind to Spinoza’s appeals to experience (see also E2a4 and especially E5p23s),36 but Spinoza’s commitment to experience is undercut in the previous sentence: “it is sufficient for me here to have shown one through which I can explain it as if I had shown it through its true cause [per veram causam]” (E2p17s; emphasis added). For although Spinoza writes to Tschirnhaus that “the Cartesian principles of natural things are useless, not to say absurd” (Ep. 81), in this limited respect Spinoza shows himself a true Cartesian for whom causal explanations of nature are always merely hypothetical.37 That is, for a Cartesian nature is often too complex to be knowable by human inquirers.38

(p. 165) This skepticism about empirical knowledge of nature is an important feature of a famous passage in a letter to Henry Oldenburg in which Spinoza illustrates man’s lack of natural knowledge by comparing man’s situation to that of a worm living in blood:

Let us conceive now, if you please, that there is a little worm living in the blood … it would live in this blood as we do in this part of the universe, and would consider each particle of the blood as a whole, not as a part. Nor could it know how all the parts of the blood are restrained by the universal nature of the blood, and compelled to adapt themselves to one another, as the universal nature of the blood requires, so that they harmonize with one another in a certain way. (Ep. 32)

Although in context Spinoza is describing a kind of natural harmony, which underwrites his general conservation law, we could label this passage (with a nod to Kant) as the Copernican revolution in Spinoza’s thought. Man lives on a small globe within an “absolutely infinite” universe, whose “parts are restrained in infinite ways by this nature of the infinite power, and compelled to undergo infinitely many variations” (Ep. 32). In context it is clear that Spinoza compares man’s situation to a worm to ridicule final causes that ascribe intentions to God.39 In addition to the argument in this letter, Spinoza’s attack on using the inductive and empirical argument from design is well-known from E1app. This suggests that, first, when he writes in TTP’s chapter on miracles, first, that “we cannot know [God’s] providence from miracles, but that all these things are far better perceived from the fixed and immutable order of nature” (TTP 6/G 3:82; see also TTP 6/G 3:86); and, second, when he identifies God’s providence with “the order of nature” (TTP 6/G 3:89), he is not showing all his cards on the subject. Even in TTP his true views are not hard to discern, however. Near the end of the book, he remarks with delicious irony, “no traces of divine justice are found except where the just rule; otherwise (to repeat again the words of Solomon [Eccl. 9:2]), we see that the same outcome happens both to the just and the unjust, the pure and the impure. Indeed, this has caused doubts about divine providence among a great many people who thought that God reigns directly over men and directs the whole of nature to their use” (TTP 19/G 3:231).40

Besides pertaining to the controversy over final causes, Spinoza’s Ep. 32 to Oldenburg is significant because it shows that Spinoza thinks it is hopeless to expect to discover true causes in nature. As he writes, “For as to the means whereby the parts [of nature] are really associated, and each part agrees with its whole, I told you in my former letter that I am in ignorance.” In context, Spinoza offers two connected arguments: first an epistemic argument—our partial vision of the universe is too (p. 166) limiting; second, an ontological-methodological argument—to know the cause of any event on Earth we need to be able to situate that cause in the infinite chain of causes. To know anything we need to know everything.41 (This is not to deny that we do know something; apparently we know that there is a universal harmony of some sorts.)

This skepticism about the very possibility of empirical knowledge of nature runs through Spinoza’s books.42 For example, in TTP he writes, “We are completely ignorant of the very order and connection of things, i.e., of how things are really ordered and connected” (TTP 4/G 3:58), and in E1app, “Since those things we can easily imagine are especially pleasing to us, men prefer order to confusion, as if order were anything in nature more than a relation to our imagination” (G 2:82). Here is a final example: “it would be impossible for human infirmity to follow up the series of particular mutable things, both on account their multitude, surpassing all calculation, and on account of the infinitely diverse circumstances surrounding one and the same thing, any one of which may be the cause of its existence or non-existence” (TdIE §100). Of course, this skepticism is compatible with a view that allows useful, local claims to be made with some probable confidence.

So this section has revealed six aspects of Spinoza’s philosophy of nature:

  1. (i) Spinoza does not use empirical knowledge as a touchstone for true, rational knowledge. Rather, in the manner of Descartes, intellectual conception is a constraint on how deliverances of the senses can be interpreted.43 This is not to deny that like Descartes Spinoza has some utility for empirical evidence. In TTP he explains, for example, that “experience cannot give any clear knowledge of these things, or teach what God is, and how he supports and directs all things, and how he takes care of men, still it can teach and enlighten men enough to imprint obedience and devotion on their hearts” (TTP 5/G 3:77–78).

  2. (ii) Spinoza associates empirical evidence with the imagination, that is, the first kind of knowledge (E2p40s2).44 (p. 167)

  3. (iii) Spinoza’s tendency to associate empirical evidence with imagination offers some evidence for his reservations about empirical evidence; it should incline us to be more cautious about thinking of Spinoza as a fellow traveler of modern science. Of course, mechanical philosophers could also be mistrustful of empirical approaches to nature, so this is by no means conclusive.

  4. (iv) Spinoza is quite adamant that we should not read the Ethics as providing foundations for a mechanics. Even in the so-called physical interlude it is not Spinoza’s “intention to deal expressly with body”; he admits he could “have explained and demonstrated these things more fully” (E2le7s).

  5. (v) Spinoza doubts that we can ever know true causes in nature.

  6. (vi) Spinoza repeatedly claims that we are ignorant of nature (e.g., “If they say that there are infinitely many things which we cannot perceive, I reply that we cannot reach them by any thought … ” (E2p49s)), and given that we need to know everything to know anything there are good grounds to treat Spinoza as a skeptic about empirical knowledge of nature.

1C.  Definitions and Essences

In TTP Spinoza is rather terse on what he means by definition.45 But the second part of TdIE is devoted to articulating the meaning and method of discovery of definitions (TdIE §49 and §94). Here I focus only on Spinoza’s views on definitions for things other than substance. A “perfect” definition explicates (explicare) “the inmost essence of a thing” (TdIE §95). Moreover, when it comes to noneternal, created things (creata res) “the perfect definitions must include the proximate cause” of the thing, and it must show how “all the things’ properties [proprietates] can be deduced from the definition.”46 Such a definition must somehow exclude other entities (see also E1p8s2) so that only one thing and all its properties are deduced from the definition. We can summarize these requirements as saying that a true definition gives a recipe from which one constructs (or in Hobbesian terms, generates) a thing with all its properties and only that thing. That is, Spinoza focuses on something like what Hobbes would call a genetic definition.47 Moreover, it’s not merely a how-possible construction but also an actualizing (p. 168) construction: “every definition must be affirmative,” (TdIE §96 and E2p4d; on necessity, see E1a3).

So for Spinoza inductive empirical inquiry is aimed at the discovery of what entities are and how they are put together. On Spinoza’s account entities have essences from which all the properties follow and which require a cause in order to exist; except for the causa sui, this cause is in some sense not part of the essence.48 This leaves two important issues unresolved. First, does Spinoza wish to distinguish between what we would call the intrinsic and extrinsic properties of thing? We can infer from some of Spinoza’s remarks that definitions deal only with intrinsic properties. For example, he writes, “No definition implies or expresses a certain number of individuals, inasmuch as it expresses nothing beyond the nature of the thing defined. For instance, the definition of a triangle expresses nothing beyond the actual nature of a triangle: it does not imply any fixed number of triangles” (E1p8s2). It seems extrinsic properties are excluded from a definition.49

Second, because the proximate cause is itself an effect and part of an infinite chain (E1p28) an infinite regress threatens; for Spinoza “the knowledge of an effect depends on and involves the knowledge of a cause” (E1a4). Some commentators have wished to avoid this conclusion by denying that the “proximate cause cannot be an array of concrete causes of its existence.”50 The only way to prevent an infinite regress is to claim that the proximate cause is God, who is, after all, “absolutely the first cause” (E1p16c3). But this argument saddles Spinoza with the implausible claim that every definition must explicitly include God.51 If Spinoza had intended this he could have claimed that about definitions, but he does not do so. In addition, his practice reveals otherwise: in the Ethics nearly all of Spinoza’s definitions do not include God. Moreover, E1p28s and its corollaries imply that God is the proximate cause of all eternal and infinite things and deny that God is a remote cause of singular things. By contrast, the existence of an infinite regress fits nicely with and reinforces the skeptical reading previously developed. It follows that no complete true definition exists capturing the actual essences of finite things (see also E1p33s1 and TTP 4/G 3:58 already quoted).

(p. 169) One might think that there is a tension. For it looks as if on the reading developed here a thing’s proximate causes involve extrinsic properties. Given that definitions include proximate causes this seems to violate the requirement that definitions exclude extrinsic properties. The apparent paradox looks like this: (i) proximate causes are contained in the definition of a thing; (ii) proximate causes involve extrinsic properties; (iii) definitions include essences; but (iv) extrinsic properties are not involved in the essence.

We can avoid paradox by noting an important peculiarity of Spinoza’s project. By way of clarification, we must first note that Spinoza thinks much of what philosophers tend to say and think about what are often called “universal” notions is confused (E2p40s1). So we must be cautious here. Nevertheless, a way out of the apparent paradox is to realize that according to Spinoza essences are not located in space and time. This will take some explaining because there is a tendency to treat definitions and essences as corresponding to each other, but the key point is that Spinozistic definitions bring together two sources of being: essences and proximate causes. These do not (to speak metaphorically) occupy the same realm of being.52

I quote one of Spinoza’s most complicated passages: “God is not only the cause of things’ beginning to exist, but also of their persevering in existing that is, in scholastic terms, God is the cause of the being of things (essendi rerum). For … so long as we attend to their essence, we shall find that it involves neither existence nor duration. So their essence can be the cause neither of their existence nor of their duration, but only God, to whose nature alone it pertains to exist” (E1p24c). This doctrine states that to say that God is the cause of things as they are in themselves is not to speak of their existence in space and time (see also E5p29s). Rather, it means that God is the (efficient) cause of their being or essence (see also E1p25).53 So whatever essences of things are they are not, as such, located in space and time. This reading of E1p24c fits with other, more straightforward Spinozistic doctrine. For example, in the explanation to E1d8 Spinoza asserts that the essence of a thing is an eternal truth (and this seems crucial to the arguments at the end of Part V of the Ethics). And we have already seen in Ep. 10 to Simon de Vries that eternal truths “do not have any place outside the mind.”

When we are dealing with definitions of finite things, we bring together two ways of being and of knowing. First, it involves knowledge of essence and its properties—this is purely intellectual knowledge.54 Recall from the letter to de Vries that the empirical world provides us no information about essences. However, while intellectual conception can provide knowledge of particular things, these are not—for lack of a better term—instantiated materially in space and time. In more modern vocabulary, this is knowledge of types not tokens. So a straightforward way to avoid any tension is to claim (p. 170) that a thing’s proximate causes are to be found at this level. (See TdIE §101: “although these fixed and eternal things are singular, nevertheless, because of their presence everywhere, and most extensive power, they will be to us like universals, or genera of the definitions of singular, changeable things, and the proximate causes of all things.”) This is not as strange as it sounds; every human has as a proximate cause, for instance, a father and a mother. Second, it involves our incomplete empirical knowledge of the machinery of the world, where individual things are to be located in space and time and where we can find the matter for their instantiation (as tokens) and the “external” causes for their destruction (E3p4). Spinoza explicitly distinguishes these two levels when he notes that “by the series of causes and of real beings I do not here understand the series of singular, changeable things, but only the series of fixed and eternal things,” (TdIE §100; the remainder of the paragraph is also highly relevant).55 Definitions bring together essences, which are fixed and eternal, and proximate causes, which belong to the world of changeable things.

The main thing that is left ambiguous in Spinoza’s account is the epistemic relationship between these two levels. (The ontic relationship—what is the process by which essences get instantiated?—is also not easy to fathom but need not concern us here.) In particular, in TTP Spinoza seems to insist that our knowledge of definitions is in some sense inductive. Yet we have not merely seen how the intellect’s knowledge is a firm constraint on the deliverances of the senses but also that knowledge of essences is not derived through the senses. The best way to make sense of the status of empirical inquiry in Spinoza is threefold. First, it can help the mind focus its attention on essences. Second, it helps uncover partial explanations. This is illustrated by Ep. 41, in which Spinoza describes an experiment he performed with two others to establish water pressure in a tube. He concludes his discussion: “The three of us were busy, to the best of our abilities, and we performed the trial with more precise results than before, but not as precise as I would have wished. Nevertheless, I got enough indication to draw something of a conclusion in this matter.” And third, empirical inquiry alerts us to potential problems in the supposed deliverances of the intellect. In Ep. 26, Spinoza reports his conversations with Christiaan Huygens about recent discoveries with microscopes and telescopes.56 Among these are empirical refutations of Descartes’s views about Saturn (and its ring—unknown to Descartes, who interpreted it as satellites of Saturn). Although Spinoza ridicules Descartes, he does not treat this as a falsification of Descartes’s principles. Rather, in context it’s clear that Spinoza thought that Descartes misapplied his own principles. So we cannot use this example as an instance where empirical claims can correct principles derived from intellectual conception (nor can we use it as evidence for the claim (p. 171) that Spinoza accepted Descartes’s principles and merely objected to Descartes’s articulation of them).

What these examples from the letters to Jelles and Oldenburg teach us is that Spinoza did not think one could always unambiguously derive knowledge of the actual machinery of nature from first principles. In his exchanges with Oldenburg about Boyle’s experiments, Spinoza makes clear that by themselves the results of experiments can be analyzed and explained in various ways. Anticipating Duhem, Spinoza argues that Boyle has to add hypotheses (about invisible particles and their natures) to infer his favored interpretations of these experiments. An experiment is useless in proving something fundamental about nature (Ep. 6).57 It yields mere probabilities.58

In a letter to Hugo Boxel we can read in a simple way the upshot of Spinoza’s methodology: “In practical life we are compelled to follow what is most probable; in speculative thought we are compelled to follow truth. A man would perish of hunger and thirst, if he refused to eat or drink, till he had obtained positive proof that food and drink would be good for him. But in philosophic reflection this is not so. On the contrary, we must take care not to admit as true anything, which is only probable. For when one falsity has been let in, infinite others follow” (Ep. 55). That is, Spinoza distinguishes sharply between useful, empirical knowledge, which is always merely probable, and durable and certain theoretical (or as I argue later) rational self-knowledge. It’s the latter that is unabashedly promoted by Spinoza: “In life, therefore, it is especially useful to perfect, as far as we can, our intellect, or reason. In this one thing consists man’s highest happiness, or blessedness … So, the ultimate end of the man who is led by reason, that is, his highest desire, by which he strives to moderate all the others, is that by which he is led to conceive adequately both himself and all things which can fall under his understanding” (E4app4).

2.  Spinoza’s Criticism of Mathematical Science

This section argues that despite contrary appearances Spinoza was very critical of applying mathematics and measurement in understanding nature. I identify different strands and arguments that explain his concern. Moreover, I argue that from the fact that he (rhetorically) deploys a geometric method in his presentation of his views, we cannot infer anything about a privileged epistemic status for geometry (or mathematics more generally).

(p. 172) 2A.  The Letter on the Infinite

Spinoza’s low expectations about the application of mathematics to nature will surprise many who think that the Ethicsmos geometricus must imply that Spinoza is a kind of modern mathematical physicist.59 Moreover, Spinoza’s library holdings at his death reveal a keen student of mathematics—among other things, he owns six volumes of Diophantus, a copy of the Mathematical Works of Vieta (1646 edition), and Van Schooten’s Geometry, the leading textbook of Descartes’s geometry.60 In this section I argue that Spinoza is critical of both the very idea that the book of nature is written in the language of mathematics as well as the very possibility that measurement can be a guide toward truth about nature—both commitments were central to the developing practice of physico-mathematics.

Before I turn to Spinoza, I make a fivefold, strictly heuristic, and simplified distinction to capture attitudes toward the relationship between mathematics and nature among leading thinkers among the “new” philosophers in the first half of the seventeenth century. First, Galileo called mathematics the language of the book of nature.61 Second, Descartes insisted that extension has geometric properties.62 Third, Newton (post-Spinoza) claims that geometry just is the art of measurement.63 Fourth, Hobbes thought that mathematics is conventional and thus based on proper (but not arbitrary) definitions (by wise legislators).64 All these views imply that to know geometric truths means one has (privileged) access to claims about nature even if the epistemic status of geometry and mechanics differ. Moreover, fifth, starting from Galileo (especially via Huygens), theory-mediated measurement is privileged in the new science of motion. In practice, there are a lot of blended positions. There is no evidence that Spinoza accepts the first three attitudes; I argue there is good reason to believe he rejected these.65 There is strong evidence he rejects the fifth. I now argue these points by articulating the details of Spinoza’s views on the relationship between mathematics and knowledge of nature.

(p. 173) Here I focus on a remarkable passage in a justifiably famous letter to Lodewijk Meyer called the “Letter on the Infinite”:

From the fact that when we conceive quantity abstracted from substance and separate duration from the way it flows from eternal things, we can determine them as we please, there arise time and measure—time to determine duration and measure to determine quantity in such a way that, so far as possible, we imagine them easily. Again, from the fact that we separate the affections of substance from substance itself and reduce them to classes so that as far as possible we imagine them easily, arises number, by which we determine [these affections of substance].

You can see clearly from what I have said that measure, time, and number are nothing but modes of thinking, or rather, of imagining. So it is no wonder that all those who have striven to understand the course of nature by such notions—which in addition have been badly understood—have so marvelously entangled themselves that in the end they have not been able to untangle themselves without breaking through everything and admitting even the most absurd absurdities. For since there are many things which we cannot at all grasp by the imagination, but only by the intellect (such as substance, eternity, etc.), if someone strives to explain such things by notions of this kind, which are only aids of the imagination, he will accomplish nothing more than if he takes pains to go mad with his imagination. (Ep. 12/G 4:56–57)

The letter may have had a fruitful afterlife in nineteenth-century history of mathematics, but that does not concern us here.66 First, the passage presupposes a distinction between (i) knowing things as imagining—confusingly to modern readers, in Spinoza’s vocabulary this can be a form of abstraction—and (ii) knowing things by way of the understanding, or rationally.67 So it fits nicely with views we have already attributed to Spinoza (recall his treatment of the vacuum). (For warnings against abstraction, see TdIE §93.) Second, in Spinoza’s complicated epistemology, knowing things by abstraction is less adequate than knowing them by the understanding (E1p15s). For Spinoza to imagine something does not always mean it is false. But it can never yield adequate knowledge (see E2p49s).68

Third, it follows from the text and these two points that Spinoza thinks that the use of measure and number do not reveal to us how substance and eternity are. Because measure and number are crucial in applying mathematics to nature one can say without hesitation that Spinoza thinks mathematics does not help us get at how reality really is but only at how we imagine it.69 This does not mean that Spinoza thinks mathematics is (p. 174) fundamentally unreliable; presumably he thinks that geometry provides a reliable form of topic-neutral inference. (Later I recount more uses for geometry in Spinoza.) He has, rather, reservations about the applicability of mathematics. Number and measure do not reveal ultimate reality (e.g., the nature of substance, eternity); Spinoza also seems to have thought that nature has more conceivable parts than numbers we can assign to it (see Ep. 83).

Fourth, we should note how broad Spinoza’s condemnation is. He is ruling out the science of motion as a privileged form of knowledge, for without “time and measure,” assigning velocities, places, and trajectories is impossible.

Fifth, of course, one wishes to know what Spinoza’s arguments are for his views. From this letter to Meyer we can infer that according to Spinoza when things are “determined” mathematically, we focus on things that have infinite number of relations with (infinite) other things; by applying measure we create what we may call a limitation of some part of the whole that is (without complete knowledge of the whole) arbitrary.70 That is, when we use measure to “carve out” a part of nature (i.e., a mode) for close study we somehow are in no position to have adequate knowledge of the whole and, thus, of it (the mode). Recall from the treatment of the worm analogy that for Spinoza to know anything we must know everything. Spinoza seems to connect that principle with the limitations on the application of mathematics.

To be clear, this does not imply that Spinoza thinks applying mathematics to nature is without use,71 for, sixth, there is a hint of what he has in mind in Ep. 6, where he implies that without experimental testing one can infinitely divide bodies and calculate forces. Spinoza does not elaborate.

2B.  Applied Mathematics and Measurement as Inadequate Knowledge

Now the passage in the quoted Letter to Meyer is not an isolated occurrence moment in Spinoza’s writings. The distinction between inadequate imaginative knowledge (or belief) and adequate rational knowledge is very Spinozistic; as we saw in Spinoza’s treatment of the vacuum in the Ethics on the rational side is undifferentiated substance, while inadequate abstraction is presupposed to locate things in (measurable) time and space. Besides the passage about the impossibility of a vacuum, there are other examples in the Ethics: “we can have only a quite inadequate knowledge of the duration of things (E2p31), and we determine their times of existing only by the imagination (E2p44s), which is not equally affected by the image of a present thing and the image of a future (p. 175) one … and the judgment we make concerning the order of things and the connection of causes, so that we may be able to determine what in the present is good or evil for us, is imaginary, rather than real” (E4p62s). The main point of this passage is unconnected to the application of mathematics (although the passage reinforces it), but the skepticism about adequate knowledge of the causal structure of nature is unmistakable; when we locate things at a time and place we are always in the realm of the imagination.

Spinoza’s reservations about the application of mathematics to establish measure and time are especially striking in light of historical context. The towering figure of Dutch natural philosophy of the period, Christiaan Huygens (1629–1695), was well-known to Spinoza—they lived near each other during the 1660s through 1666, when Huygens moved to Paris. From Spinoza’s correspondence we can infer that they spoke not merely about their shared interest in lens cutting and optics but also about many other topics. One of Huygens’s main intellectual breakthroughs in developing Galileo’s science of motion was to provide a mathematical analysis of isochronous (pendulum) clocks. Moreover, by having a mathematical analysis of the properties of a pendulum available, he was able to establish the speed of falling bodies, and thus the pull of gravity, with remarkable precision (up to four significant figures) and accuracy. Huygens’s insight consisted of realizing that the pendulum itself can be both timekeeper and an experimental measure; the pendulum is a falling body, so the swinging pendulum contains within itself the theory-mediated measure of gravity.72 Spinoza knew of some of Huygens’s work on the pendulum (Ep. 30A; Oldenburg repeatedly asked him about it), and he owned Huygens’s 1673 masterpiece Horologium Oscillitarium.73 While Huygens would not deny that such measures contained a margin of error, Spinoza’s remarks suggest that he thought that even in principle mathematically designed clocks are unable to ever reveal adequate knowledge of the duration of things. More important, even if they were somehow error free they would still not capture the essential nature of things.74 Clocks do not reveal the causes of why things go in and out of existence.

We can infer from his scattered remarks on the subject that Spinoza links the mathematical approach to nature with a kind of piecemeal understanding of it. If we read Spinoza as a Cartesian this would be baffling because by linking extension to geometry Descartes thought he had made secure knowledge of the machinery of nature possible. Because Spinoza is not shy about naming Descartes as the target of his criticisms, here he is best read as offering an informed interpretation of the new physico-mathematics pursued by Galileo and Huygens, who—to simplify—studied, for example, the pendulum as a closed system. Eighteenth-century Newtonians would praise the incremental, piecemeal approach to nature and would single out Spinoza’s demand for systematicity as a form of intellectual hubris.75 Of course, Spinoza’s point generalizes to any incremental, piecemeal method that assumes a closed system for the sake of analysis.

(p. 176) 2C.  Mathematical Overconfidence

The Appendix to Part I of the Ethics is widely read and noted because of its attack on final causes. From our post-Darwinian perspective it is tempting to read Spinoza as “one of us,” especially because Spinoza goes well beyond Descartes’s cautious rejection of final causes in physics. Spinoza was deeply suspicious of final causes in general, and this much is accepted widely among scholars even by those who insist that Spinoza’s psychology or his treatment of the conatus doctrine still smuggles in teleological explanation. Spinoza rejected general final causes such as promoted by proponents of the argument from design or God’s providence (e.g., Boyle and after Spinoza’s death, Newton) and local final causes in the explanation of mechanism. An example of a local final cause is the Epicurean conception of gravity, where the body just knows which way is down. This doctrine is presupposed by Boyle in his exchange with Spinoza (see Ep. 11).76 Spinoza ridicules final causes as the product of anthropocentric fears and aspiration, “those things we can easily imagine are especially pleasing to us, men prefer order to confusion, as if order were anything in things [ordinem in rebus] more than a relation to our imagination” (E1app/G 2:82). It is noteworthy how broad Spinoza’s attack is here—he seems to be claiming that all perception of order in nature is really a projection. In light of the skeptical strain we have already identified this should not surprise us.77 Of course, this is not to deny that, perhaps, from God’s perspective there is order.

Some readers may be tempted to understand another remark in the Appendix to Part I of the Ethics as Spinoza’s endorsement of mathematics. It occurs in the context of explaining why despite the fact that the attribution of final causation is a natural fallacy. Spinoza writes, “if mathematics, which is concerned not with ends, but only with essences and properties of figures, had not shown men another standard of truth. And besides mathematics, we can assign other causes also (which it is unnecessary to enumerate here), which were able to bring it about that men would notice common prejudices and be led to true knowledge of things” (G 2:79–80).

Spinoza is making four points in this passage. First, of course, the invention of mathematics allowed humanity to develop a standard of truth other than one based on final causes. But note, second, that Spinoza explicitly denies that the invention of mathematics was a necessary condition to develop epistemic criteria that allow one to escape the (p. 177) reign of final causes.78 This is not exactly a ringing endorsement of mathematics. Third, he tantalizes the reader with unnamed, “other causes” that could have had the same beneficial outcome. This deflates any argument for the special status of mathematics in Spinoza’s thought based on this passage.

Spinoza tends to associate mathematical figures with abstraction, so it would be surprising if he would praise mathematics highly. In fact, just a few lines down in the same Appendix Spinoza turns to thinly veiled criticism of mathematics: “there are men lunatic enough to believe, that even God himself takes pleasure in harmony; indeed there are philosophers who have persuaded themselves that the motions of the heavens produce a harmony” (G 2:82). This is a barb at Kepler’s and young Huygens’s astronomical Platonism (his 1659 Systema saturnium appeals to harmonic principles). While mathematics is not named, the perception of harmony is a consequence of the search for mathematical order in nature. Rather than being a reliable guide to nature as it really is, mathematics promotes the tendency to project harmony or beauty in nature where there is none.

Fourth, while few would defend the claim that Spinoza thinks experiments lead to fundamental natural knowledge, many think it is obvious that according to Spinoza mathematics helps us gain knowledge about physical bodies.79 Spinoza understands mathematics as the discovery of essences and properties of figures, that is, the constructability of geometric figures. Thus, for Spinoza mathematical knowledge is a model for the content not so much of natural knowledge but of the form of knowledge; mathematics teaches us the importance of essences and properties.

2D.  Mos Geometricus

I want to forestall a general objection to the reading presented in this chapter; it is based on Spinoza’s style of presentation in the Ethics.80 The geometric method has tempted many commentators into thinking that Spinoza was a friend of the developing sciences of the period. Moreover, in the Preface to Part III of the Ethics, Spinoza writes, “Therefore, I shall treat the nature and powers of the affects, and the power of the mind over them, by the same method by which, in the preceding parts, I treated God and the Mind, and I shall consider human actions and appetites just as if it were a question of lines, planes, and bodies” (G 2:138). Spinoza was clearly willing to present his views about human affairs as well as nonhuman things in the language of geometry. (p. 178) To be clear he is not translating his views about God or human affairs in the language of geometry—rather, he is creating a mode of presentation that is analogous to the language of geometry.

Also, it is worth realizing that much of the Ethics is not composed more geometrico: this includes E1app and E4app; E3pref, E4pref, and E5pref; the definitions of the passions; and the many long commentaries attached to propositions. And new definitions are introduced in later parts; Spinoza leaves it ambiguous if these need to be applied retrospectively (this is a nontrivial matter with E2d2). Moreover, at several occasions Spinoza makes it clear that the Ethics is not a purely deductive work but has holistic qualities. For example, he writes, “For the present, I cannot explain these matters more clearly” (E2p7s), and “Here, no doubt readers will come to a halt, and think of many things which will give them pause. For this reason I ask them to continue on with me slowly, step by step, and to make no judgment on these matters until they have read through them all” (E2p11s, emphasis added). See also E2p49s with its explicit forward reference to part V and the brief Preface to Part II, which alludes to closing lines of the Ethics. This is all very different from how Spinoza presents how we ought to think about knowledge of nature.81

Furthermore, the choice for presenting his views more geometrico appears to be informed by substantive views about the kind of authorial persona Spinoza wishes to convey as well as his views about education. The most detailed account in Spinoza’s corpus of the virtues of the pedagogical virtues of the Mos Geometricus is supplied by Lodewijk Meyer in his Preface to DPP: it is said to be the safest and most secure method for teaching knowledge. In particular, this mode of presentation offers the student hope and security. The student learns how to become rational step by step. Rather than sowing doubts, it offers intellectual security. In this manner the student is elevated above the “vulgar.” (TTP 5/G 3:77 emphasizes that a few people wish to be taught in this fashion; see also TTP 13/G 3:167; see also the Second Set of Objections to Descartes’s Meditations.) This fits nicely with Spinoza’s view that the true teacher avoids discipleship but teaches to become an independent thinker (TTP 1/G 3:16, especially the note added in Spinoza’s hand).82 Spinoza appears to believe that this mode of presentation directs attention away from the author and to the work’s content (see TTP 7/G 3:111).

Meyer also claims that the geometric method helps avoid polemics. At the start of the Political Treatise, Spinoza calls attention to the value neutrality and independence that is supposed to be conveyed by this mode of presentation:

In turning my attention to political theory it was not my purpose to suggest anything that is novel or unheard of, but only to demonstrate by sure and conclusive reasoning such things as are in closest agreement with practice, deducing them from human nature as it really is. And in order to enquire into matters relevant to this branch of (p. 179) knowledge in the same unfettered spirit as is habitually shown in mathematical studies, I have taken great care not to deride, bewail, or execrate human actions, but to understand them. So I have regarded human emotions such as love, hatred, anger, envy, pride, pity, and other agitations of the mind not as vices of human nature but as properties pertaining to it in the same way as heat, cold, storm, thunder, and such pertain to the nature of the atmosphere. (TP 1.4)

3.  The Metaphysical Foundations of Natural Philosophy

In this section I treat Spinoza’s second and third kind of knowledge. I review these in light of their significance, if any, regarding Spinoza’s views on natural science. I start, however, with an analysis of Spinoza’s views on motion. Many of Spinoza’s critics and not a few of his friends discerned serious problems with Spinoza’s treatment of motion. I focus primarily on to what degree, if any, Spinoza’s metaphysics lends itself to offering a foundation for mechanics. By contrasting Spinoza to Descartes, Huygens, and Newton, this will naturally lead to a revisionary discussion of the role of common notions. Their role is primarily as a stepping-stone to the third kind of knowledge. My treatment of intuition emphasizes its orientation toward self-knowledge.

3A.  Motion and Conservation

In an important letter to Oldenburg, Spinoza formulates a general conservation law: “the relations between motion and rest in the sum total of them, that is [bodies], in the whole universe, remain unchanged” (Ep. 32). From the Ethics we discern that Spinoza’s arguments in favor of this conservation law are conceptual, not empirical (E2le7s; via the proofs of E2le7, E2le4, and E2le1 Spinoza refers to E1p15s and the arguments for the denial of a vacuum in nature).

It is unclear if Spinoza has a compelling argument for his general conservation law because he admits that if he had wished to “deal expressly with body, [he] ought to have explained and demonstrated these things more fully” (E2le7s). It appears that his general conservation principle is founded on three deeply anchored Spinozistic principles. First, there is only one substance; from this it follows, second, that everything is systematically connected to each other, and third, that (despite the heterogeneity of the appearances) matter is homogenous.83 With these principles one can guarantee that, given movement, the relationship between motion and rest must remain the same. But (p. 180) these three principles do not guarantee that there is motion at all. Moreover, without an analysis of motion it is unclear why the appearances ought to be interpreted as in motion. (At one point Spinoza is clearly concerned about the issue because he has an extensive treatment of Zeno’s paradox of motion in DPP2p6s.) Because Spinoza rejects Descartes’s God, who sets the whole chain of motion in motion (E1p28; see also Ep. 73), there is an apparent lacuna in Spinoza’s system.84 It is not anachronistic to raise these issues because in Spinoza’s criticism of Descartes he signaled awareness of the significance of the issues: “for matter at rest, as it is in itself, will continue at rest, and will only be determined to motion by some more powerful external cause; for this reason I have not hesitated on a former occasion to affirm, that the Cartesian principles of natural things are useless, not to say absurd” (Ep. 83).

A number of related issues here are worth exploring and distinguishing. First, Spinoza takes it as axiomatic that there is motion and rest in the world (E2a1’). Moreover, he seems to accept a distinction between “absolute” and (presumably) merely apparent motion (E2le2). So his rejection of the anti-Copernican position (recall section 1A), a rejection requiring that apparent motion can be distinguished from real motion, can be accommodated by his metaphysics. Also, motion is one of the individuation conditions of simple bodies (E2le3, especially the demonstration— “rest, quickness and slowness” are the other criteria). In fact, a compound individual is an entity (or nature) that maintains the same ratio of motion and rest among its parts (E2le5).85 So motion plays a crucial role in Spinoza’s fundamental metaphysics.86

Yet, second, as even his admirer, Toland, noticed, Spinoza never defines what motion is.87 Because Spinoza is so critical of Descartes on these matters, we cannot simply assume that he has taken over Descartes’s definitions.88 In fact, it is not easy to imagine (p. 181) what anything but a heuristic analysis of motion in Spinoza would look like. It would require introducing spatial and temporal notions into one’s reflection on infinite extension; establishing velocity would require measurement. As we seen, all of these operations involve having an inadequate conception of reality according to Spinoza.

Third, even if we grant that we can supply Spinoza with a fruitful conception of motion, it is not obvious he could have a compelling story about the source of motion. In Ep. 83, Spinoza writes that “matter at rest … will persevere in its rest, and will not be set in motion unless by a more powerful external cause.” Given that Spinoza’s God is immanent (E1p18; Ep. 73), there is no “external” cause that sets the infinite chain of matter in motion. Matter at (absolute) rest generates no motion; therefore, this implies that according to Spinoza there must be motion in the universe from the “infinite start” (E1p28).89 Now Spinoza offers sufficient reason for this at E1p16: “From the necessity of the divine nature [who has absolutely infinite attributes by E1d6] there must follow infinitely many things in infinitely many modes.”90 From a human vantage point this does not offer sufficient explanation. Even if granted that there must be infinitely many things in infinitely many modes, this does not seem to explain why it is a feature of the necessary system that there is motion. The infinitely many things in infinitely many modes are all (to speak informally) possible things, and one might wonder whether motion is impossible. It certainly leaves the impression that the origin of motion is unaccounted for. Spinoza’s critics starting with Henry More in the Confutatio (1678) were quick to notice the problem.

Fourth, Samuel Clarke noticed a peculiar feature of Spinoza’s system. Spinoza treats the universe as a whole as an individual in which the proportion of motion and rest remains the same (E2le4, E21e7). But maintaining this proportion is compatible with the quantity of motion varying in the universe. Clarke draws a very important observation from this: “there might possibly have been originally more or less motion in the universe than there actually was.”91 Thus, the proportion of motion and rest can remain the same while the quantity of motion can change. For example, if some parts move faster to accommodate the faster motion in other parts, then the proportion may remain the same even though the quantity of motion increases. That there might possibly have been more or less motion in the universe violates both the PSR as well as Spinoza’s claim that “things could have been produced by God in no other way” (E1p33).92 Of course, in light (p. 182) of Spinoza’s necessitarianism, Clarke’s point is merely a conceptual possibility, but that is sufficient for his purposes.

Long before he completed the Ethics, Spinoza informs Blyenbergh: “I have never thought about the work on Descartes, nor given any further heed to it, since it has been translated into Dutch” (Ep. 38). We can only regret that Spinoza never wrote a treatise about mechanics.

3B.  Common Notions (and Laws of Motion/Thought)

Recall that in TTP, Spinoza writes that “in examining natural things we strive, before all else, to investigate the things which are most universal and common to the whole of nature—viz., motion and rest, and their laws and rules, which nature always observes and through which it continuously acts and from these we proceed gradually to other less universal things” (TTP 7/G 3:102).93 Spinoza alludes here to an important concept in his epistemology: so-called common notions. In handwritten note six that Spinoza added to TTP he makes clear that common notions are stepping-stones to adequate knowledge of God—in particular, that God “exists necessarily, and is everywhere” and that God’s nature is presupposed in all things we conceive” (G 3:252–53).94 While in this chapter I have been emphasizing a skeptical strain in Spinoza, this concept seems to offer a robust route to adequate knowledge of the second kind (E2p40s2). In particular, one might think that I have given far too much attention to the unattainability of the third kind of knowledge and the limitations of the first kind of knowledge within Spinoza while downplaying the presence of the second kind. Yet Spinoza writes that human minds contain adequate ideas (E3p1d) and refers to E2p40s2, so common notions seem available to all.

Now, given that motion and rest, and their laws and rules are said to be common notions, it is no surprise that many readers think that Spinoza is here asserting that we can have adequate knowledge of (Cartesian-style) physics.95 But we have already seen that Spinoza’s use of laws intends to convey the unchanging and deterministic nature of nature rather than any entity that figures into, say, a science of motion.

The fact that rules of motion and rest are common notions is more important. This does echo a Cartesian program of scientific explanation with laws of motion and/or rules of collision. But the similarity with Descartes is superficial, as reflection on the nature of common notions reveals. They are structural features that all modes within an attribute share: E2p38c appeals to E2le2, which in turns follows from the definition of a body (E2d1). Therefore, just as there are common notions of modes of extension, so (p. 183) there must be common notions of modes of thought. To put the point metaphorically, the economy of thought is just as rule governed as the economy of extensional nature for Spinoza (a most un-Cartesian thought). Of course, given parallelism (and E2p39, more explicitly), this means that these laws and rules, whatever their content, are going to have a high degree of generality and relatively little specificity. So what are common notions?96

First, common notions are about qualitative not quantitative properties of extension. The manner or magnitude of such properties is extrinsic and thus is not a common notion. This becomes clear by reflection on how Spinoza characterizes common notions: common notions are qualities that all bodies share regardless of their state (see, especially, E2p38–9; to be clear, Spinoza does not use qualities to describe common notions). Second, these properties do not just have a high degree of generality—they are common to all bodies (E2le2, cited in E2p38c)—but the manner in which they are present within each and all bodies is also equal (E2p39d).97 The best way to make sense of common notions is, thus, to suggest that they are intrinsic properties of modes within an attribute (in Spinozistic terms they share an affection) and that they reflect the peculiar modal qualities of such a mode: for example, all bodies are equally capable of motion and of rest, of moving slower and quicker (E2le2), capable of being an efficient cause, of codetermining and terminating other bodies (E1d2, E1p28, E2le3).

This last feature certainly draws Spinoza very close to Descartes’s laws of motion. For, example, Descartes’s first law states “that each thing, as far as is in its power, always remains in the same state; and that consequently, when it is once moved, it always continues to move” (PP 2.37) a claim that is fairly close to Spinoza’s corollary to E2le3: “a body in motion moves until it is determined by another body to rest; and that a body at rest also remains at rest until it is determined to motion by another.” It is fair to say that Spinoza makes explicit what Descartes intended: that bodies are causes of each other’s motion and rest. So Spinoza is Cartesian insofar that he accepts Descartes’s general program by which observed changes in motion (or rest) encourage the search for other bodies that caused these changes.

Even so, there are interesting differences: Cartesian “inertial” motion is a consequence of the state-preserving power inherent in each thing, whereas Spinoza offers no such consequence relation in his lemma.98 A more important difference is that Spinoza (p. 184) lacks the equivalent of Descartes’s second law of motion: “all movement is, of itself, along straight lines” (PP 2.39). This is no trivial matter. It means that Spinozistic inertial motion can take any “shape” (e.g., circular, rotational, zigzagging). Intuitively, Spinoza’s move makes sense: from the point of view of (say) eternity, it is not obvious why states (of motion) need to be preserved along a straight line. This requirement seems to introduce an arbitrary directionality and even geometry into mode continuation and preservation. Given that Spinozistic laws of extension and laws of thought are, in some important sense, the same, such directionality would probably make a mockery of the very possibility of finding rules of thought that are identical to rules of extension (and any other attribute). It is also by no means obvious how the directionality requirement can be derived or justified metaphysically.99 While we do not tend to connect rectilinear motion with final causes, one can easily imagine that to Spinoza attributing some such “knowledge” of direction to a moving body must have reeked of superstition (akin to Epicurean innate gravity).

The downside of Spinoza’s approach is that it is very hard to see how in the absence of a detectable body, B, acting as cause(s) on some body, A, we can ever say about some moving body, A, that it was in inertial motion or not. Given that E2a1” explicitly allows that the way bodies move each other (as causes) is potentially heterogeneous, the epistemic complications of using Spinoza’s axioms and laws as foundations for a science of motion are only increased. So commentators that attribute to Spinoza the idea that his common notions enter into his science of motion saddle Spinoza with a decidedly unpromising physical science.

Now it is possible that Spinoza did not recognize any of the problems I have indicated. (Note, by the way, that I am not relying on later developments in physics.) It is possible, of course, that even after Christian Huygens published Horologium oscillatorium sive de motu pendularium (1673), which articulated how Galilean principles could be developed into a science of motion, Spinoza was unwilling to drop his alternative approach. But given that Spinoza has so many criticisms of mathematical physics, a more obvious interpretation presents itself. Spinozistic common notions are not the foundation of a Spinozistic physical science (analogous to Cartesian, Huygensian, Leibnizian, Newtonian) (mechanics). Rather, they capture secure knowledge of the modal qualities that are intrinsic to all modes of an attribute. This is the meaning of E5p4: “there is no affection of the body of which we cannot form a clear and distinct concept.”100 That (p. 185) is, common notions provide us knowledge of the nature of bodies (E2p16). This is not nothing, of course, and such common notions are significant because with Spinozistic metaphysics they provide hope that access to third kind of knowledge is available to mere mortals (E2p47s).

3C.  Conception of Essences

As we have seen, for Spinoza knowledge is about intellectual conception of eternal essences.101 The third and highest kind of knowing “proceeds from an adequate idea of the formal essence of certain attributes of God to the adequate knowledge of the [formal] essence of things” (E2p40s2). Spinoza’s meaning here has puzzled generations of readers. The example that accompanies it does not help explain what he means by “formal essence of certain attributes” or by “formal essence of things.” In Part V of the Ethics Spinoza writes about this third kind of knowledge of (formal) essences: it is “eternal” (E5p31; see especially its demonstration). While building on this proof, Spinoza refers to E1a3, which insists on causal necessity (E5p33d). The necessity of causation is commonplace in the seventeenth century. Spinoza is, perhaps, a bit unusual in not accepting any exceptions to natural necessity either for God or for mankind. He rejects the conception of “man in Nature as a dominion within a dominion” (E3pref/G 2:137).

The third kind of knowledge is the source of “the greatest satisfaction of the mind: that is, … joy” (E5p32d). So it would be congenial to learn what it is knowledge of and how we obtain it. On the first point, I just quoted the mysterious passage (“proceeds from an adequate idea of the formal essence of certain attributes of God to the adequate knowledge of the [formal] essence of things”) from E2p40s2; on the second point, Spinoza writes in E5p31, “the third kind of knowledge depends on the mind, as on a formal cause, insofar as the mind itself is eternal.”

According to Spinoza the source of the third kind of knowledge is within the mind itself (e.g., E5p30). According to the proof of E5p31 this is the case because to be a formal cause is synonymous with being an adequate cause of the third kind of knowledge. The proof refers to the first definition of Part III of the Ethics. To be an adequate cause means that one acts from one’s nature, that is, one is acting from reason (E4p23–26) or that one understands something as necessary (E2p44). Helpfully, Spinoza clearly points out on four occasions that this does not involve what we tend to call knowledge of empirical nature. Beyond the (p. 186) cited passage about the denial of the vacuum, I would like to call attention to the demonstration to the second corollary of E2p44, the whole of E2p45, and most clearly E5p29s, which I quote: “We conceive things as actual in two ways: either insofar as we conceive them to exist in relation to a certain time and place, or insofar as we conceive them to be contained in God and to follow from the necessity of the divine nature. But the things we conceive in this second way as true, or real, we conceive under a species of eternity, and their ideas involve the eternal and infinite essence of God (as we have shown in E2p45&s).”

With fully adequate knowledge, the knower and the known object coincide and dissolve each other as distinct beings—this is why the mind becomes eternal (E5p40). What’s crucial for present purposes is that this third kind of (self) knowledge is contrasted with knowing something in relation to a certain time and place. To assign time and place to modes one must, as we saw in the passage about the denial of the vacuum (E1p15s), use abstraction or imagination to discern determinate and separable regions of pure quantity (see also E2p44c1s and E2p45s).

In a letter to Johannes Bouwmeester, Spinoza summarizes these complicated matters in simple fashion. I quote (in my own translation): “all clear and distinct ideas which we conceive can only be caused by other clear and distinct ideas, which are in us, and do not permit another cause outside of us. From this it follows that the clear and distinct ideas, which we conceive, only depend on our nature and her determined and fixed laws, that is to say, our absolute power, and not on chance, that is to say, from causes which, howsoever obeying determined and fixed laws, are unknown to us and outside our nature and power” (Ep. 37; see also E5p40s).

All of this implies that according to Spinoza when we conceive of things at a place and time we are dealing with our lack of power and thus imperfect, fallible knowledge. We learn from the opening pages of the unfinished Treatise on the Emendation of the Intellect that unsettled things cannot make us happy. It is no surprise, then, that Spinoza is quite critical of mathematical natural science. His epistemic concerns fit with his moral aims. It is therefore a mistake to understand Spinoza as a fellow traveler of the scientific revolution.

When it comes to having adequate ideas, then, we are not perceiving things outside of us in spatial and temporal places or locations (i.e., things we are inclined to call knowledge of nature), but we are in possession of a special kind of self-knowledge. For Spinoza, godly substance is knowable (“with great difficulty”; E1p15s) through ourselves (E5p30). Of course, for Spinoza knowledge of mechanics is not a primary goal; for him physics is subservient to “knowledge of the human mind and its highest blessedness” (see E2pref).


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(1) As should be clear from what follows, this chapter is primarily devoted to Spinoza’s views on mechanics and what we would call philosophy of science. (For useful comments on the many ways science can be used in context of Spinoza’s life and works, see Gabbey, “Spinoza’s Natural Science.”) I remain largely silent on Spinoza’s contributions to the human and interpretive sciences. I defer to future research a thorough analysis of Spinoza through an optical lens.

When I started researching this paper I was very much a novice in Spinozistic matters. I am very grateful to Michael Della Rocca for his encouragement. On earlier drafts of this chapter and related works I have been privileged to receive detailed and thorough comments from a true community of scholars, including from Alex Douglas, Alan Gabbey, Don Garrett, Helen Hattab, Bryce Huebner, Michael Le Buffe, Charlie Huenemann, Monte Johnson, Mogens Laerke, Steve Nadler, Alison Peterman, Sam Rickless, Don Rutherford, Noa Shein, Tad Schmaltz, Piet Steenbakkers, Kevin von Duuglas-Ittu, and, of course, Michael Della Rocca. I suspect all the folk just named will be disappointed that I stubbornly resisted adjusting the text in light of some of their most critical comments.

Finally, I should note the existence of two as of yet unpublished dissertations by Peterman, “Spinoza and the Metaphysics of Finite Bodies,” and Douglas, “Spinoza’s Vindication of Philosophy,” who both explore Spinoza’s critical distance from the way seventeenth-century Cartesian and Galilean physical sciences are being developed and offer many independent lines of argument in support of the main tenor of this chapter (although they should not be implicated in my mistakes).

(2) This claim is treated as uncontroversial by Morgan in his edition of The Essential Spinoza, p. 216. It can be found as well in Garber, “Descartes and Spinoza,” p. 64; Nadler, Spinoza: A Life, pp. 192–93.

(3) An intellectual biography of Spinoza would have to trace Spinoza’s early embrace of the mechanical philosophy as demonstrated, especially, by Descartes (with Ep. 6 to Oldenburg as high point) to his mature rejection of Descartes’s philosophy of nature (explicitly in Ep. 81 to Tschirnhaus).

(4) See, for example, the lovely research by Kevin von Duuglas-Ittu, a very creative independent scholar, at “Deciphering Spinoza’s Optical Letters”; for more of his research on Spinoza, see

(5) Jonas, “Parallelism and Complementarity”; see also the very influential Bennett, A Study, p. 92.

(6) See, for example, Viljanen, “Field Metaphysic.” Viljanen offers a brilliant defense of Bennett’s field metaphysic and sees Spinoza as solving the metaphysics required for motion.

(7) As recently as 1985 Petry felt secure in attributing two anonymous pieces, Stelkonstige reeckening van den regenboog and Reeckening van kanssen, published anonymously in 1687 to Spinoza in his edition of Spinoza’s Algebraic calculation of the rainbow; &, Calculation of chances. As Petry notes in his editorial introduction, there is considerable evidence that Spinoza composed and probably burned a short treatise on the rainbow, but there is no evidence that he ever composed a treatment on probability. The attribution of these pieces to Spinoza has been decisively refuted in De Vet, “Was Spinoza de auteur”; De Vet shows Salomon Dierquens is the most likely author in Salomon Dierquens.”

(8) Jonathan Israel is quite aware of the contrast between, say, Newton and Locke (the emblematic figures of so-called moderate Enlightenment) and Spinoza (the inspiration of the so-called radical Enlightenment), but he still closely identifies Spinoza with science, the scientific revolution and even “mathematical logic;” see Israel, Radical Enlightenment, p. 242. The whole of this chapter is meant as a challenge to Israel’s views.

(12) References to the Theological-Political Treatise are to the recently published Curley edition.

(15) TTP 7/G 3:103 is also ambiguous: “Once this universal teaching of Scripture is rightly known, we must proceed next to other, less universal things, which nevertheless concern how we ordinarily conduct our lives and which flow from this universal teaching like streams. For example, all the particular external actions of true virtue, which can only be put to work on a given occasion.”

(16) Here I ignore a complication: in TdIE §98, Spinoza is talking of knowledge of essences, not definitions. I explain the relationship between essences and definitions later.

(18) See DPP2p14–7. See Gabbey, “Spinoza’s Natural Science,” pp. 156–68, for discussion.

(19) Some recent commentators have identified infinite modes with laws of nature; for discussion, see my treatment later of common notions.

(20) One reason to be skeptical about treating the mature Spinoza as a mechanical philosopher is that he never seems to point to (geometric) shapes of bodies as important explanatory principles.

(21) This claim should not be overemphasized. In the terminology (and nominalism) of TdIE (unlike that of the Ethics) “fixed and eternal” things and “changeable” things are “singular.” I thank Don Garrett for pressing this point.

(22) Nature is a notoriously slippery concept. Unless otherwise specified in this chapter I mean to be referring to the subject matters that are the object of contemporary natural sciences in the broad sense (Spinozistic natura naturata).

(23) For a contrary view see Marshall, “Adequacy and Innateness.” Nadler treats common notions as innate ideas in Spinoza’s Ethics, p. 175. Cf. James, “Spinoza on the Politics.”

(24) This is not said as evidence of Spinoza not being a mechanical philosopher (many of whom—Bacon, Boyle, Huygens—were very empirical). Even Descartes engages in important empirical research; see, for example, Buchwald, “Descartes’s Experimental Journey.”

(25) Cf. TTP 4/G 3:76.

(26) Klever, John Locke. (1632–1704). Vermomde en Miskende Spinozist—Een Vergelijkende Studie.

(27) There is a further complication because Spinoza implies that modes are also eternal truths but that he avoids calling them by that name to avoid confusion. This letter provides evidence for idealist-friendly interpretations of Spinoza.

(29) See Bell, Christiaan Huygens, p. 164.

(30) An unpublished paper by Helen Hattab indicates that Spinoza may be relying on Gorleaus.

(33) See Bennett, A Study, p. 99.

(34) For the exact details of the manuscript he probably possessed, see Spruit and Totaro, The Vatican Manuscript.

(37) Savan, “Spinoza: Scientist,” p. 114. My whole chapter is deeply indebted to Savan’s pioneering treatment.

(38) This is not to deny that at the end of Principles Descartes offers an inference to the best explanation and consilience arguments to claim that his system has moral certainty.

(39) von Duuglas-Ittu has called attention to Kircher’s Subterranean World, which had been mentioned by Oldenburg in the previous letter that Spinoza is answering, as a source for Spinoza’s image; Kircher had announced that worms could be found in the blood of fever victims (von Duuglas-Ittu cites Ruestow as his source). See von Duuglas-Ittu, “A Worm in Cheese.”

(40) While Van Velthuysen may have misunderstood Spinoza on some matters, surely he got this right! (Spinoza does not challenge this aspect of Van Velthuysen’s reading in his response.) See Ep. 42 Van Velthuysen to Ostens.

(41) First, on some readings of E1a4 it might be taken to support the claim that according to Spinoza to know anything we need to know everything. But, as Della Rocca pointed out to me, all it can be made to say is the weaker claim that to know something one must know its cause and the infinitely many prior causes.

Second, at E3p1d, Spinoza quite clearly asserts that all human minds contain some adequate ideas. Regardless of the origin of these ideas, Spinoza does not claim that adequate ideas are “active” in everybody. The reference to E2p40s1 makes clear that Spinoza is thinking of common notions here.

(43) I have used the locution intellectual conception rather than intellectual perception because as Piet Steenbakkers first pointed out to me Spinoza sometimes tends to associate perception with the first kind of knowledge. By intellectual conception I mean to convey adequate cognition by the intellect.

(44) Cf. Bennett, A Study, p. 24, who argues that Spinoza makes room for “experiential non vaga” or controlled evidence, but acknowledges there is little textual evidence for this. I thank Alex Douglas for the pointer.

(45) The only useful remark is, “A law which depends on a necessity of nature is one which follows necessarily from the very nature or definition of a thing…For example, that all bodies, when they strike against other lesser bodies, lose as much of their motion as they communicate to the other bodies is a universal law of all bodies, which follows from a necessity of nature” (TTP 4/G 3:57–8). Cf. Descartes’s third law of motion: “a body, upon coming in contact with a stronger one, loses none of its motion; but that, upon coming in contact with a weaker one, it loses as much as it transfers to that weaker body” (PP 2.40). But I will rely largely on TdIE because it offers more and clearer content on the matter.

(46) For the significance of proprietates as opposed to propria, see Melamed, “Spinoza’s Metaphysics of Substance.”

(48) In “some sense” is deliberately vague. Except for substance, no thing’s essence fully contains its own cause.

(49) The model seems to be the way formal causes work as sources of geometric construction in sixteenth- and seventeenth-century geometry. See Mancosu, Philosophy of Mathematics. On this matter, I am very indebted to discussion with Karolina Hübner.

(51) The problem is avoided if every definition is tacitly thought to include God (for example, if the essence of a thing is related to an essence of an attribute of God (as in the third kind of knowledge) or if the cause(s) described lead back to God, as Steve Nadler has suggested to me. E2p10 implies, however, that substance does not belong to the essence of man, so I see no reason to think that God should figure in the definition of a human or any other mode (even if nothing can be or be conceived without substance). However, I doubt there is evidence for the suggestion that there are two kinds of definitions (a partial one of particulars available to us, which we might label imaginative, and a complete one, which we might label rational), which was suggested to me by Tad Schmaltz.

(52) That is, it will involve a rejection or redescription of the second premise in the apparent paradox.

(53) This also means that God’s immanent causation is not about the cause of singular things, that is, finite, determinate entities (located ‘in’ space and time), but about the cause of the essences of things (cf. E1p24c). So on my reading, the “in” part of immanence should not be understood spatially.

(54) In private communication, Alison Peterman has usefully pointed out that this is a kind of counterfactual knowledge of what an essence would necessarily cause in the absence of other things.

(55) TdIE §100 is often taken to claim that the fixed and eternal things just are proximate causes, but the reading developed here relies on the thought that would be more accurate to say that they are to us like proximate causes.

(56) For interesting context and material on the astronomical issues discussed, see von Duuglas-Ittu, “What Spinoza and Huygens Would Have Seen.”

(57) It is unfortunate that we have no evidence for Spinoza’s specialist reaction to Newton’s early optical experiments and the controversy with Huygens they generated.

(58) My interpretation has been anticipated by Savan, “Spinoza: Scientist”; Savan also has a lovely discussion of Spinoza’s use of models in empirical inquiry.

(59) A careless reading of the closing lines of the Preface to Part III may reinforce the first impression.

(61) See Galileo, The Assayer.

(62) PP 2.23 and 2.64.

(63) “Geometry is founded in mechanical practice, and is nothing but that part of universal mechanics which accurately proposes and demonstrates the art of measuring.” Author’s Preface to the Principia. For more sophisticated treatment see Guicciardini, Isaac Newton on Mathematical Certainty.

(64) This is a controversial reading of Hobbes. But see Hanson, “Reconsidering Hobbes’s Conventionalism,” p. 642; Jesseph, Squaring the Circle, pp. 199–201.

(65) Obviously, my claim that Spinoza rejects the second is most controversial. Later, I quote from Spinoza’s “Letter on the Infinite” to support the position. There is also indirect support for this claim. First, when Spinoza famously defends his application of the geometric method to “human vice and folly,” he is making clear that he is deploying a topic neutral method; this suggests he severs any special link between geometry and substance. Of course, this observation does not preclude the possibility that the features of extension are captured by geometry. Second, throughout his mature writings, Spinoza is very critical of Descartes’s account of natural philosophy in general and Descartes’s conception of extension in particular (see especially Ep. 83).

(66) Bussotti and Tapp, “The Influence.” I thank Kevin von Duuglas-Ittu for calling my attention to it. See also Savan, “Spinoza: Scientist,” p. 96, on Frege.

(67) It is tempting to think of the imagination and the understanding as different faculties or mental capacities, but this cannot be right. Besides Spinoza’s rejection of faculty language, it is clear that Spinoza thinks of imagining as a mode of thinking. Yet according to Spinoza imagining is about bodies and thus is not fully real or adequate thought. It is not my charge to explain this.

(68) De Dijn, Spinoza: The Way to Wisdom, p. 150, reads TdIE as claiming a distinction between intellect and imagination, “or the true idea from the fictitious, false.”

(70) Another way to approach this issue is through Spinoza’s remark in the Ethics that “being finite is really, in part, a negation” (E1p8s1). Negation can never lead to complete knowledge.

(71) Spinoza also appears to think that mathematicians are confused about the nature of number, but that does not concern us here.

(74) The allusion to Kantianism is deliberate, but this is not the occasion to pursue a historical argument.

(76) Some readers attributed the Epicurean notion of gravity to Newton, who was eager to distance himself from it—see his famous “Letter to Bentley.”

(77) A very important proposition for much recent interpretation of Spinoza, E2p7, which underwrites what many people call Spinoza’s parallelism, may be thought to contradict the view I am articulating. In the context of this handbook I cannot articulate an interpretation that does justice to the complexity of the proposition, its corollary, and very long scholium, which Spinoza ends with a disarming, “I cannot for the present explain my meaning more clearly.” Spinoza’s remark suggests that no straightforward reading of the proposition is forthcoming.

(78) Spinoza is frustratingly silent on what alternative causes might be available, but I suspect that he believes the rule of law—with security of life and liberty—reduces terrors that promote search for final causes.

(79) See, for example, a footnote by a Dutch translator, Henri Krop, of the Ethics (note 46 to Part II, p. 539).

(80) In this section I go against the consensus view on Spinoza’s deployment of the geometric method. For a very learned and clear introduction, see Steenbakkers, “The Geometrical Order.”

(81) My comments here do not touch another use for the geometric method, namely, to signal that truth has been arrived at apodictically. I thank Alan Gabbey for his critical comments.

(82) For Spinoza’s concerns about discipleship, see Cooper, “Freedom of Speech.”

(83) This is a core commitment of most seventeenth-century “new” philosophers. Newton finally abandons it by the second edition of the Principia.

(84) DPP is more Cartesian on this score (see DPP2p11–2).

(85) In the Preface to Part II of KV, a ratio of 1:3 is mentioned (G 1:52), but it is unclear if it is Spinoza’s position or an editorial addition. From the vantage point of this chapter, I offer two points: first, if Spinoza did once believe that there were fundamental equations in nature, he seems to have thought better of it as he matured; second, this ratio echoes, as von Duuglas-Ittu points out, Descartes’s sixth collision rule—interestingly, the very one that Spinoza explicitly disavows in the letter fragment to Oldenburg. See von Duuglas-Ittu, “The ‘Corporeal Equation’ of 1:3.”

(86) This is denied by Bennett, A Study, p. 106. Accordingly, Bennett distinguishes between “motion” at the “most basic level,” where it captures a way of speaking about “alterations in space” and a more “ordinary sense” (pp. 106–7); shortly thereafter we learn that “Spinoza did not become perfectly clear about the difference between the ground floor and the next level up.” Bennett’s interpretation is far removed from the text.

(87) See Toland, Letters to Serena, especially chapter 4. I thank Dennis Des Chene for calling my attention to it. (It is unclear if motion even can be defined if it is a mode of extension. As Noa Shein pointed out to me it is hard to say in terms of what it could be defined.) Nevertheless, in chapter 5 Toland defends the (Spinozistic) doctrine of activity as essential to matter. Nadler, Spinoza’s Ethics, p. 196, fn. 7, has pointed to the conatus doctrine as evidence that Spinoza rejects the passivity of matter, and that it has innate active powers. Diderot seems to have read Spinoza this way (see Wolfe, “Rethinking Empiricism”; Wolfe, “Endowed Molecules”).

(88) A good thing, too, because as Newton demonstrated most clearly in his unpublished tract, “De Gravitatione,” these definitions are defective in generating an even moderately useful treatment of motion.

(89) Samuel Clarke thinks that E1p33 and E2le3 contradict each other on the origin of motion. It is not obvious what Clarke has in mind (see Clarke, A Demonstration, part VIII, p. 45). These are only in contradiction if God (as producer of motion) and God as the infinite chain of causes are in no sense identical (and this is not obvious one way or another, although it would require equating substance with an infinite mode). Even so, given that E1p33 treats God as natura naturans, while at E2le3 we seem to be in the realm of natura naturata, Clarke is probably onto a significant problem here. For more discussion, see Schliesser, “Newton and Spinoza.”

(90) “Ex necessitate divinae naturae infinita infinitis modis (hoc est, omnia, quae sub intellectum infinitum cadere possunt) sequi debent.”

(91) Clarke, A Demonstration, Part VIII, p. 45.

(92) A way to save Spinoza’s adherence to the PSR is to distinguish between a strong version of the PSR, which governs all of nature, and a weak version, which governs only those entities that have full reality (eternal truths). Even “absolute” motion would not have full reality. But this is not the place to pursue such a controversial matter.

(93) This section draws on material that I have first articulated in Schliesser, “Angels and Philosophers.”

(96) In the secondary literature one often finds the answer: “infinite modes” (and these are often thought to be scientific laws of nature). The evidence for this claim is remarkably thin (it requires reading E1p21–3 in light of Ep. 83). But even if one grants the equation among infinite modes, common notions, and laws of nature, this does not license the further inference that common notions are the building blocks of a science of motion, or mechanics.

(97) One might think that in E2p39d Spinoza is discussing a more restricted class of common notions, namely, those that are common only to the human body and the bodies with which it usually interacts. (By contrast the common notions of E2p38 would be universal.) But I see no other evidence to think that Spinoza thinks that there are bodies different in kind such that they would not share in the common notions of the bodies that can affect our bodies.

(98) When Spinoza does state his conatus doctrine later at E3p6–7 it is traced back to Spinoza’s understanding of the expression doctrine (E1p25c), God’s power (E1p34), what it means to be an “essence” (E1p36), and a “determinate nature” (E1p29). Motion is strikingly absent in motivating or explaining the conatus doctrine.

(99) This is, I think, why E2a2’ ’ which does offer nontrivial directionality constraints on the way collisions proceed, is offered as an additional axiom. It is very hard to see what justifies treating it as a common notion. It is also very different from Descartes’s third law of motion, which is supposed to govern collision (and from which the particular rules of collision are claimed to be derived). This is not the place to offer a substantive interpretation of it, but Spinoza may have thought it follows from some kind of least action principle. I thank John Grey, Rodolfo Garau, and Alison Peterman for discussion.

(100) The demonstration of E5p4 reads as follows: “Those things which are common to all can only be conceived adequately (by E2p38), and so (by E2p12 and E2le2) there is no affection of the body of which we cannot form some clear and distinct concept.” The inference makes perfect sense if common notions pick out the knowable modal qualities that are intrinsic to all modes of an attribute; if extrinsic qualities are thought to be included in an “affection of the body” then Spinoza’s inference begs the question. This interpretation fits how Spinoza implies that the non-affections of the human body are not known to the mind at E2p24.

(101) While this is well-known in Spinoza scholarship, it gets ignored by folks when they assimilate Spinoza to the mechanical philosophy. This neglect may be partly motivated by disquiet about the fact that this is used as a complaint by a very hostile source, Albert Burgh (see Ep. 67).