A.D. Maidanskii

The Reform of Logic
in Descartes’s and Spinoza’s Works

Russian Studies in Philosophy,
vol. 37, no. 2, Fall 1998, pp. 25-44.

Sooner or later there comes a time in the history of a science when it pauses from dealing with its innumerable special problems and returns to the study of first principles and the foundations that delimit its particular field of inquiry. The result is usually a radical revision of a number of established basic ideas, the discovery of some hitherto unknown dimension in the field, and the emergence of an appropriate paradigm for investigating the field. It was the fate of logic to be one of the first fields (second only to astronomy) to undergo such a reformation, which lasted more than two centuries. Its instigators were René Descartes and Benedict Spinoza.

It was mainly the achievements of mathematics that provided the impetus for this reform. Since the times of Pythagoras and Plato, philosophers had endeavored to penetrate the secret of the extraordinary clarity and certainty of mathematical truths. Quite a few rules of Aristotelian logic, undoubtedly, were based on the techniques of mathematical construction, which were widespread at the time. In particular, the logical method of “proof from the opposite” was used by the Pythagoreans, to demonstrate the irrationality of the square root of two, long before the Stagirite described it in his Analytics. 1

Descartes fully shared this notion of the ancients: “Mathematics [25] accustoms the mind to recognize the truth, because it is in mathematics that examples of correct reasoning, which you will find nowhere else, are to be found. Accordingly, the man who has once accustomed his mind to mathematical reasoning will have a mind that is well equipped for the investigation of other truths, since reasoning is exactly the same in every subject.” 2 The search for a universal “method of reasoning” that is “the same in every subject” and that enables us to extract the truth from any object is the business of logic. At the same time, it was perfectly clear to Descartes that the rules, schemata, and laws of traditional logic were of no help at all in understanding the actual course of reasoning in mathematics and theoretical thought in general. Guided by such a logic alone, a theoretician would be simply incapable of discovering any new truth.

Descartes said: “I observed in regard to logic that syllogisms and most techniques are of less use for learning things than for explaining to others the things one already knows. ... This really applies not so much to logic ... but to dialectic, which teaches us how to hold forth on all subjects. ... It diverts us from the actual nature of the thing itself.” 3

Accordingly, real logic has no right to set aside with indifference the “nature of things” and to concern itself only with the forms of thought that are common to all objects indiscriminately. Indeed, a scientist needs not so much the dialectical art of “reasoning about everything” as the ability to understand the nature of concrete individual things. That is where logic is supposed to help him.

But how is the logician to know how to steer thought correctly toward the nature of things? Descartes answers that logic could best learn this from mathematics. Mathematics is the complete logical method in action. However, in mathematics the logical method appears dressed in numbers and figures that hide its true face even from the mathematicians who use it. By freeing mathematical thought from numbers and figures and studying the mechanics of its action in pure form, we could, Descartes assumes, acquire a universal method enabling us to discover the truth in any subject whatsoever. Descartes calls the science that treats of such a method mathesis universalis, a name that goes back to the times of Proclus Diadochus or even earlier. “Such a science should contain the primary rudiments of human reason, and its province ought to extend to the eliciting of true results in every object.” 4 Essentially, mathesis universalis is nothing other than [26] logic, but a logic so different from traditional logic that Descartes avoids using the term “logic” so as not to obscure its distinctiveness. (Later, the expression mathesis universalis disappears from his vocabulary, and Descartes calls his logic simply “method”).

The distinctiveness of this discipline which, in Descartes’s mind, is to become the instrument of theoretical thought, lies in its orientation to the general order of nature. Descartes holds that there exists in nature a single order, which embraces such diverse things as numbers and figures, stars and sounds, everything that is accessible to the human mind. Mathesis universalis studies this order in pure form and sees to it that the order of thought in all domains of knowledge corresponds to the general order of nature. In this sense it is the “method that teaches us how to follow the true order.” 5

Here is the essence of the reform in logic that Descartes initiated. Traditional logic only saw to it that a verbal or any other symbolic expression of thought corresponded to itself (hence the law of identity and the ban on contradiction become its supreme first principles) and entrusted the task of investigating the general order of nature to metaphysics. Metaphysics in turn made active use in its construction of the formal methods and recommendations of traditional logic, so that in the end the order of nature appeared to metaphysicians in the image of a pyramid of genus and species abstractions like Porphyry’s famous categorial tree.

Setting about to construct his own logic, Descartes soon qualified this view, declaring that the true order of nature has nothing in common with what is taught in the accepted textbooks on metaphysics. The “chief secret” of his method of thought, according to Descartes, is that “all facts can be arranged in a certain series, not indeed in the sense of being referred to some ontological genus such as the categories employed by philosophers in their classification, but insofar as certain truths can be known from others...” 6 The order of things in nature must be construed in the categories of reason and consequences, cause and effect, and not in accordance with the formal scheme of genus and species, Descartes never tires of repeating.

According to the key postulate of mathesis universalis, in any series of mutually related things, there is always one thing that serves as the reason or cause of all other things in that series. Descartes calls these things-reasons “absolute” things. All other things, “relative” things, are modes of the “absolute” things and are understood only by being [27] “correlated with absolute things and derived from them.” 7 Usually, the main difficulty is to discover this first reason, and its certainty determines the certainty of our understanding of the entire series of relative things. Hence an absolute thing must always be marked as such prior to all others, according to rule 6 “for the direction of the mind.” An absolute thing forms a “pure and simple nature” of which all other things of a given series partake to some degree. They are united not by the formal similarity of some attributes, but by their common origin and genesis, the source of which is the absolute thing. The latter is a kind of gene from which all more concrete structures grow in strictly determined sequence, and any newborn thing brings with it some “relation” proper to it alone. The method teaches “that these [natures] should all be distinguished and their correlative connections and natural order so observed that we may be able by traversing all the intermediate steps to proceed from the most remote to that which is in the highest degree absolute.” 8

Descartes warns us in rule 12 that the logical order of the knowledge of individual things does not correspond to the order of their real existence. Every individual thing acquires its existence from some other individual thing and in turn imparts existence to a sequence of things. Reason is interested not so much in describing this order that vanishes into infinity or the innumerable circumstances for the existence of things as in the reason why a thing exists in this and not in some other way. This cannot be understood if we do not know the “natures” lying at the basis of individual things and defining the law of their existence. The universal “natures” discovered by reason do not exist separately, but reveal themselves only through individual things, Descartes explains. 9 This doctrine of absolute things, in which the clear and simple nature of what exists is directly manifested, is the prototype of the dialectical idea of the concrete universal that “exists as its own prime species” (Hegel), that is, shows itself in the form of an individual thing that constitutes the cause or the reason of the existence of other individual things. 10

Descartes borrowed the described method of raising the individual to the universal from the “algebra of the ancients” (in particular from the Alexandrian mathematicians Pappus and Diophantus) and developed it by extending it to the entire theory of magnitudes. This led him to the discovery of analytic geometry. By taking a specific unit as the basis for an entire series of magnitudes, the mathematician is able to [28] represent the relationship of all kinds of magnitudes, regardless of whether they are numbers or figures, in the form of a proportion in which the unit is the “common measure” for all terms of the proportion. (In every concrete case one can take as a unit “either one of the already given magnitudes or any other magnitude, which will then be the common measure for all the others”; 11 in spatial intuition it is not difficult to present a unit as a point, a line, or a square.) With such a proportion at his disposal a mathematician can derive analytically any unknown from any known magnitude.

For Descartes deriving new definitions on the basis of existing ones is the only correct schema of action for theoretical thought in any realm of knowledge. There is always a specific proportion, more or less evident and complex, between any scientific discovery and a set of previously accumulated knowledge. Descartes was the first to make this internal proportionality of theoretical thought an object of logical inquiry. It is no accident that Descartes’s favorite metaphor for true knowledge is a chain in which each new link or idea is so firmly connected to the preceding one that we “easily see how the first and the last are linked together.” 12 Similarly, true thought for Descartes is an infinite sequence of ideas flowing one from the other in an order that is no less rigorous than that which exists in a series of continuously proportional magnitudes. But where is the “absolute ground” of this entire order, the proto-idea of thought, so to speak? Until this is known, not one of the ideas at the mind’s disposal, including those that relate to the theory of method, can be considered absolutely certain. Hence the search for the proto-idea becomes the decisive test of the Cartesian method, for the latter’s reliability can be confirmed only by grounding it in the most fundamental idea of true thought.

The ancient philosophers almost unanimously considered the supreme idea of reason to be the thought of the first principle of all being. But when philosophers had ascertained finally that among all possible opinions concerning first principles no opinion was accepted as incontestable by everyone, the skeptics turned their attention to the nature of the human mind: is it not the mind that makes this dispute unresolvable? Reason is subject to some natural ailment, wrote Michel Montaigne. Like a silkworm, it is condemned to entangle itself in its own constructions and risks being suffocated by them. 13 Francis Bacon likened human reason to a mirror with a distorted surface, so that we could never be completely sure that a mental image of any object, and [29] especially of the extremely abstract material of metaphysics, is adequate to the object as such.

Descartes agreed to consider the skeptics’ hypothesis of the innate imperfection of the human mind and even assumed that some omnipotent evil genius exists who “applied all his ingenuity to lead me into error.” 14 Now the issue is not simply the nature of being as such, but its principle for human thought: we can no longer postulate and have to prove that the proto-idea of human thought coincides with the idea of the real principle of the world. Thus Descartes shifts the traditional metaphysical problem of the first principle of being to the level of the science of the method of thought, that is, logic, and logic becomes an object-related discipline dealing with the “nature of things” and striving to really guide theoretical thought rather than restricting itself to the search for rules and schemata for expressing already discovered truths.

The Cartesian method first obliges us to define the unit of thought, that is, the idea that could serve as the foundation and “universal measure” (of truth) for all other ideas accessible to the human mind. As we know, Descartes chose the reflective idea of the ego, the idea of pure self-awareness—ego sum, ego existo (I am, I exist).” For Descartes, as Martin Heidegger writes, “the truth of what is ... is weighed and measured by the forces of the ego.” 15 It may prove that Protagoras had already achieved the same thing; however, Heidegger explains that for Protagoras the ego is not yet autonomous, a “subject existing in its own right,” as in Descartes.

Looking at his own ego more closely, Descartes notes at the very outset that the ego is a finite and imperfect being. Only a finite and imperfect being is capable of doubting, erring, experiencing affects, striving for something and, in general, changing in time. But in comparison with what does the mind recognize its finiteness and imperfection? There must exist in the mind, Descartes reasons, an idea of some infinite and most perfect thing, and guided by this standard-idea, the mind judges the degree of perfection of the finite things it perceives. An idea of the finite and imperfect could not be formed except from an idea of the infinite and most perfect, says Descartes, just as one cannot construct a segment without first possessing the idea of an unfinished straight line or form the idea of a limit, which lies at the basis of the theory of mathematical analysis, without the idea of a continuous sequence tending toward infinity. In a word, the idea of the finite and [30] imperfect is derivative of the idea of the infinite and most perfect and is secondary in relation to the latter.

Descartes calls this infinite and most perfect thing God in the spirit of his times and goes on to say: “in some way I have in me the notion of the infinite earlier than the finite—to wit, the notion of God before that of myself.” 16 This shows that the certainty of the axiom “I exist” depends directly on the certainty of the perception of God. Knowledge of the existence of the ego is, therefore, not an absolute truth! As long as the truth of the idea of God is not proved, we have no warrant to affirm unambiguously that the idea of the existence of the ego is true, despite its indubitable certainty for my ego. I, a finite being, get my existence not from myself: “were I myself the author of my being, I should doubt nothing and I should desire nothing, and finally no perfection would be lacking to me ... I should thus be God.” 17 Hence the method obliges Descartes to seek the basis of the existence of a finite mind outside of the ego. It is only after discovering this reason-thing and demonstrating that the mind possesses a true idea of it that Descartes could be sure that the consequence-thing, that is, the ego, really exists.

This is confirmed by the “logico-architectonic” analysis of the text of the Meditations done by the authoritative French historian of philosophy Martial Gueroult. Gueroult says that Descartes still considered his own existence to be doubtful at the beginning of the Third Meditation (“On God—that He exists”). 18 Edwin Curley is of a similar opinion: insofar as the existence of God remains unproved in the first two meditations, the Cartesian ego sum is doubtful—doubtful in the normative sense that it merits doubt, even though it may compel belief whenever it is itself attended to.” 19

Everything indicates that the Cartesian ego is not a “subject existing in its own right,” as Heidegger describes it; the ego is a finite image of the infinite—God. But the ego, seen in abstraction from its own infinite nature, would not have sufficient power even to “weigh” the truth of its own existence at the level of concept. The mirror of the Cartesian cogito does not reflect Heidegger’s autonomous subject, “an existence closed in upon itself and aspiring to become a focal point of all being, but a special mode of infinite and most perfect—“divine”—thought. The finite mind owes its existence not to itself but to God, who is for Descartes the only “subject existing in itself insofar as God, unlike the ego and all other finite things, “does not need anything to maintain [31] his existence and, thus, in a certain sense is his own cause.” 20

There is a widespread opinion that the concept of God serves Descartes merely as an auxiliary construction to which he resorts whenever he is unable to come up with a natural explanation for a particular thing. This is partly true, but, in addition, the Cartesian concept of God—especially in relation to the existence of the ego—contains a new and extremely important meaning that predetermined the subsequent history of the science of logic.

The principal attribute of God for Descartes is absolute infinity. His proof of the existence of God thus logically sanctions operations of theoretical thought using the category of infinity, especially in mathematics. On the other hand, for Descartes it is clearly mathematical proofs that serve as a model for an a priori proof of the existence of God: analyzing the idea of God in the way a mathematician analyzes figures and numbers, Descartes finds that he knows no less clearly that “eternal existence pertains to this nature than I know that all that which I am able to demonstrate of some figure or number truly pertains to the nature of this figure or number, and therefore, ... the existence of God would pass with me as at least as certain as I have ever held the truths of mathematics (which concern only number and figures) to be.” 21

“The proof of the existence of God from the simple idea of God is called an a priori demonstration. Its first version appeared in the treatises of Anselm of Canterbury. Later Anselm was supported by Bonaventure and Duns Scotus, but the party of the opponents of the a priori argument, headed by Aquinas, proved to be more influential so that in the time of Descartes even the few scholars who were familiar with this argument did not take it seriously. In its Cartesian version the a priori argument looks as follows: God is conceived as an all-perfect being and being is the supreme perfection of a thing; consequently, actual and eternal being necessarily belong to the nature of God.

To some thinkers this argument seemed to be an utterly “sublime thought,” 22 an idea that “defined the entire subsequent development of modern philosophy,” 23 while others believed that all the effort expended in demonstrating this was wasted. 24 In any case, the a priori argument is still vigorously debated by Western philosophers and theologians. 25

Regardless of whether one accepts the a priori argument as sound, one cannot deny that it is intimately connected with a new branch of knowledge that arose soon after Descartes’s death—mathematical [32] analysis. Leibniz, one of the fathers of mathematical analysis, devoted many pages to meditations on Descartes’s a priori proof and even developed his own version of this proof. Leibniz was fond of repeating that metaphysical truths were much more prevalent in mathematics than is usually thought. The importance of philosophical proofs of the existence of an absolutely infinite reality proved to be all the greater because all attempts to give a purely mathematical grounding of differential and integral calculus (Euler, Lagrange, and many other equally illustrious scholars) were for a long time unsuccessful. Pointing to this, some thinkers who rejected the a priori argument at the same time disputed the soundness of mathematical analysis as well. In his Analyst, George Berkeley demonstrates the impossibility of a concept of a derivative or a “fluxion” on the grounds that it lies beyond the finite. Berkeley is not original here: Aristotle in his Physics already said that there is no actual infinity. Descartes made people doubt fundamentally the logical truths of this position but it was only at the end of the nineteenth and the beginning of the twentieth century, after Cantor came out with his set theory and made a number of other discoveries in mathematical logic, that the category of the infinite was finally established in its just rights as an objectively valid form of thought. Essentially, in proposing his proofs of the existence of an absolutely infinite reality, Descartes showed that logic not only could learn from mathematics, but could also contribute in the most direct manner to the development of mathematical thought. No one before, and very few thinkers since Descartes, has come up with such a convincing demonstration of the heuristic potential of the method and categories of logic. But, by accepting the a priori demonstration of the existence of God, Descartes created for himself a very difficult problem: this argument not only does not depend on the self-awareness of the ego, but by its logical form it contradicts the original postulate of Cartesian philosophy: the thinking ego is here the subject and “existence” is the predicate. The a priori argument asserts the direct opposite: “my thought is predetermined by the necessity of God’s ... existence.” 26 Here primacy is explicitly given to being. Moreover, although the mind’s concept of itself, that is, the reflective idea of the ego, is formed prior to the concept of God, God, according to Descartes, is in the nature of things prior to the ego and, as it were, recreates mind at every instant of its existence. In Descartes the sequence of the derivation of concepts proves to be the direct opposite of the order of things in nature. The [33] question arises: “if the apprehension of God is more primary than the apprehension of oneself,” then why do all of Descartes’s metaphysical treatises open with the concept of the existence of the ego? Since the idea of God, according to Descartes’s expression, is “first and original” (idea prima et praecipua 27) among all ideas, why should Descartes not begin his Meditations immediately with the a priori proof of God’s existence?

The whole issue lies in the imperfection of the human spirit, its finite and dependent nature, Descartes would most probably have replied. Undoubtedly, human reason might comprehend being more adequately by forming its own ideas in the same sequence in which God creates all that is, beginning, of course, with its own idea of God. There is only one obstacle: in addition to the idea of God, the human mind also possesses a vast mass of other ideas and images, most of which are very uncertain or at least dubious. All the ideas at man’s disposal form a complex association by intricately intermeshing and overlapping each other so that the idea of God usually is easily lost in this unsurveyable mass. Hence, before placing the idea of God at the basis of the collection of true knowledge about what is, the philosopher must firmly ascertain that no alien intellectual forms and, especially, sensible images have adulterated the idea of God. Indeed, as a consequence of mixing the idea of God with images and ideas of finite things, all possible disagreements have arisen over the conception of God and, despite the fact that the idea of God is one and the same in all people, there exists no small number of uncommonly dissimilar notions of God. “And as regards God, if my mind were not preoccupied with prejudices and if my thought did not find itself on all hands diverted by the continual pressure of sensible things, there would be nothing which I could know more immediately and more easily than Him...” 28

Descartes wants to purify the idea of God and present it in a form in which it is “innate” to the mind. The instrument with which Descartes cleanses the idea of God is the reflective idea of the ego. For this reason the latter is prior to the concept of God in the system of human knowledge. Descartes deems the ego to be an imperfect copy of God or, in other words, a finite idea of an absolutely infinite reality. “God in creating me placed this idea (the idea of God) within me to be like the workman imprinted on his work; and it was likewise not essential that the mark shall be something different from the work itself.” 29 This [34] means that ego does not simply possess the idea of God, it is the idea of God, although only an imperfect and finite idea. Hence, in pondering the nature of his own ego, the philosopher naturally arrives at the thought that there is an absolutely infinite reality: “When I reflect on myself I not only understand that I am something imperfect and dependent on another, which incessantly aspires after something which is better and greater than myself, but I also know that He on whom I depend possesses in Himself all the great things ... actually and infinitely; and that thus He is God.” 30

This means that God is nothing but an actually infinite reality. Herein, in particular, lies the “purified” (via the mind’s reflection in itself) idea of God. Thanks to the a priori argument, this idea definitively establishes itself as the primary and supreme idea of reason. “And so I very clearly recognize that the certainty and truth of all knowledge depends alone on the knowledge of the true God, insomuch that, before I knew him, I could not have perfect knowledge of any other thing.” 31

Despite the fact that Descartes was able to reconcile the reflective idea of the ego and the idea of an absolutely infinite reality, such a compromise, as it turned out later on, was not very satisfying. Spinoza proposed the task of constructing a concept of God without the help of the reflective idea of the ego and thereby to reduce the derivation sequence of “the first principles of philosophy” and the order of things in nature to a common denominator. That was also how Fichte proceeded with the difference that for him the common denominator was not the idea of God, but pure self-awareness.

Spinoza wholly supported and continued the reformist efforts in logic begun by Descartes. He too strove to make logic a discipline about objects. Although Spinoza used the word “logic” with its derivatives even more rarely than Descartes, he sets forth his views on the subject and tasks of “true logic” (as he once called his theory of the method of thought), 32 more precisely and clearly. The expression “true logic” indicates that he never confused logic as such with the traditional logic that was set out in the generally accepted manuals or in the treatises of the medieval peripatetics. In fact he was quite dismissive toward the latter. For him true logic was supposed to accomplish a [35] vital practical task—to find a method to improve the human intellect.

“In what way should the intellect be perfected ... is the subject of logic,” according to the Preface to part 5 of Ethics. 33 Spinoza draws a parallel between logic and medicine, which is concerned with the proper functioning of the body’s organs; in this regard logic is the medicine of the mind, an applied discipline, which should arm the intellect with knowledge about the laws and forms of its own work. Logic is even more useful than medicine: “Since our best part is our intellect, there is no doubt that, if we truly wish to seek our advantage, we must strive above all to improve it as much as possible, for our highest good must lie in its improvement.” 34 Accordingly, logic teaches man how to achieve the “highest good,” in other words, true logic is at the same time true ethics.

But where does Spinoza expound his Logic? Obviously, in the Treatise on the Improvement of the Understanding, although the word “logic” does not appear in the text of the treatise. Like Descartes, he uses the word “method” instead. To verify this, it suffices to compare the definition of logic in the Preface to part 5 of Ethics with what is said about method in the Treatise: “The chief part of our method is to understand as well as possible (optime intelligere) the powers of the intellect and its nature.” 35 Generally speaking, Spinoza’s logical method is a reflection of the intellect on itself for the purpose of improving it.

Logical or reflective meaning is secondary relative to knowledge about external things, and the quality of the logical method depends substantially on the quality of ideas about external things, Spinoza claims. The more perfect these ideas, the better the intellect is able to understand itself, its own nature, and its capabilities.

“Again, method must necessarily be concerned with reasoning or understanding (ratiocinatio, aut intellectio), that is, method is not the same as reasoning in the search for causes, still less is it the comprehension of the causes of things; it is the understanding what is a true idea. ... Whence it follows that Method is nothing else than reflective knowledge or the idea of an idea; and, since there can be no idea of an idea—unless an idea exists previously—there can be no method without a pre-existent idea. Hence a good method will be the one that shows us how the mind should be directed according to the standard of a given true idea.” 36

To acquire a good method of thought Spinoza recommends studying [36] concrete ideas that the mind possesses. The point is that the ideas already at hand serve as a “norm” of thought that guides the mind as it broadens the domain of its knowledge about what is. But the mind often is not aware of or is wrong about what ideas move it in the process of understanding. Yet the entire course of thought depends on which of the ideas present in the mind works in the given case as a regulator or an actual “norm” of thought. The more perfect this idea-norm, the deeper and clearer the mind’s grasp of the nature of things and all newly acquired ideas increase the mind’s optical arsenal and become in their turn mental tools (instrumenta intellectualia) with which (the intellect) acquires new powers for other intellectual work.” 37 The choice of an appropriate instrument or norm of thought is of decisive significance for the success of any concrete cognitive act. The advantage of a good method, according to Spinoza, consists of helping the mind to select a norm of thought appropriate to the given object and to follow the correct order of investigation. But it should be borne in mind that the method, or “the idea of an idea,” is something different from an idea as such and, therefore, the method cannot be used as a “norm” of thought in place of a nonreflective idea. The method in general is not a means for acquiring a single new positive idea: the method is simply a searching device or an indicator enabling one to recognize among a multitude of ideas known to the mind the idea-organon by means of which one can form an idea of the object which happens to interest the mind at the moment.

Herein lies the cardinal difference between Spinoza’s method and the formal methods of scholastic logic, as well as the later speculative methods: the former does not claim to be the absolute norm of truth, but is only “a reflective knowledge in accordance with the norm of the given true idea” (cognitio reflexiva ad datae verae ideae normam). For Spinoza, the method of thought is dictated in each particular case by a concrete idea and is not imposed on the world of ideas by the logician as of some sort of a priori (or, more simply, unidentifiable) “canon of pure reason.” The purpose of method is to disclose the heuristic potential inherent in every idea, and, says Spinoza, the cognitive capacity of the mind extends exactly as far as the “intellectual instrumentation” in the mind’s possession permits. This instrumentation includes all ideas that were acquired by the mind previously, and grows with every discovery, with every new and true idea.

Under such conditions, there can be no talk of a method that is given [37] once and for all. It is easy to suppose that Spinoza comes to deny the possibility of a universal method of knowledge which generations of logicians from Aristotle to Descartes dreamed of. Thus, Dutch historian of philosophy Wim Klever comments that, in working on the Treatise on the Improvement of the Understanding, Spinoza became ever more convinced of the impossibility of constructing a logic that was distinct from “physics,” that is, from the positive knowledge of things that exist outside the intellect and are known by it. The only true logic is the “immanent logic of human thought which directs itself.” 38 That is why Spinoza left his work on the Treatise unfinished and moved on to studying nature as such. 39

Indeed, insofar as logical knowledge is reflective and the method of thought is dictated by the concrete content of any idea, there simply cannot be a logical method that is the same for all ideas. However, this is not so. Among other “intellectual tools,” the intellect has one quite special, truly universal tool—an idea suited for deriving all possible true ideas. This is the idea of Nature (sometimes Spinoza calls nature God or substance and sometimes the most perfect being). Reflection on this idea in itself, that is, the idea of the idea of Nature, turns out to be the sought-for universal or “most perfect” method of thought. “The reflective knowledge which has for its object the most perfect Being (ens perfectissimum)” said Spinoza, “is more excellent than reflective knowledge concerning other objects; in other words, that method will be most perfect which affords the standard of the given idea of the most perfect Being whereby we may direct our mind.” 40

But what does this reflective idea of the most perfect Being consist of? The answer should come from Spinoza’s investigation of the intellect, since the knowledge of nature and properties of the intellect serves as the general foundation for any adequate reflective (= logical) knowledge, including the “idea of the idea” of the most perfect Being. “If, therefore, we wish to study the first thing of all [in particular, the idea of the most perfect Being—A.M.], then it will be necessary to supply some foundation which may guide our thoughts thither. Further, since method is reflective knowledge, the foundation which must guide our thoughts can only be the knowledge of that which constitutes the reality of truth, and the knowledge of the intellect, its properties and powers.” 41

Let us see how Spinoza forms the concept of the intellect. The first step is to distinguish somehow the intellect from all other forms of [38] thought proper to the human mind. Spinoza calls this operation “the healing and purifying of the intellect” 42 and compiles a historiola mentis, a brief description of the mind on the analogy to the “natural histories” of heat, color, wind, and so forth by Bacon of Verulam. 43 In describing the forms of perception (modi percipiendi) typical of the mind, Spinoza pays particular attention to the case “in which a thing is perceived only through its own essence or through the knowledge of its most proximate cause.” 44 He deems this perception the best and later on calls it “intellect” (the three other, inadequate forms of perception belong to “imagination”).

This definition points to a characteristic distinguishing feature of the intellect—its concentration on the causes of things. But it should be noted that there is not one word on the proximate cause of the intellect itself. Spinoza simply asserts the existence of the phenomenon of the intellect and describes it briefly to enable us to distinguish it from the phenomena of the imagination when we set about inquiring into the nature of the intellect. Then come the sections on (a) what is method; (b) how a true idea differs from a false, fictitious, and doubtful one; (c) the conditions for an adequate definition of some particular thing; and (d) how to choose the correct order of knowledge. In conclusion the author gives (e) a broad review of the characteristic features of the intellect. Now that we have an exemplary portrait of the intellect, the time has come to define its essence, its “proximate cause.”

“The definition of the intellect makes itself manifest if we pay attention to its properties, which we know clearly and distinctly. ... We must lay down some common basis from which these properties necessarily follow, so that when this is given, the properties are necessarily given and, when it is removed, they too vanish with it.” 45

The manuscript of the Treatise breaks off after this sentence. However, anyone who is familiar with the rest of Spinoza’s works will have no difficulty in inferring with a high degree of confidence that the sought-for definition would be roughly as follows: The intellect is the infinite idea of the most perfect Being (substance). The proximate cause of the intellect is thought (cogitatio), since it belongs to the nature of the most perfect Being, in other words, since this Being is a “thinking thing” (res cogitans).

We find an analogous definition, for example, in the appendix to the Short Treatise, where Spinoza writes of the existence of a “most direct [39] modification of the attribute we call thought,” namely, “an infinite idea that comprises objectively the whole of nature as it really exists in itself.” 46

The definition of the intellect as the infinite idea of substance completes the “healing” of the intellect. As the idea of substance it becomes the object of reflection in Ethics in part 1 (“On God”) and the first nine theorems of part 2 (“On the Nature and Origin of the Mind”). We should note that the concept of substance given here is not simply the idea of substance—that idea as such is, to use Spinoza’s expression, an “innate tool” (innatum instrumentum) of the mind, and is not acquired by philosophizing— but a reflective idea, “an idea of an idea” of substance. And this is precisely what Spinoza calls the “most perfect method” of thought! Accordingly, the section of Ethics that sets out the reflective idea of substance is not “physics” but rather logic, although a very particular type of logic. Its distinctive feature is its concern with the nature of things; hence, Spinoza’s logic can be described with reason as a variety of object-directed logic, 47 or, more precisely, a “natural” logic, in view of the fact that for Spinoza the object of all knowledge is, in the final analysis, Nature.

Traditional logic, of course, attached no special importance to the difference between objects of knowledge and most often simply disregarded this difference. In chapter 7 of the Short Treatise on God, Man, and His Weil-Being, Spinoza directly opposes traditional logic and “true logic,” which bases its laws on the “differences among things in their very essence,” and not on the general structure of speech. In giving the classical formula of a correct definition—via genus and species difference—Spinoza comments, “Although all logicians agree with this, I, nonetheless, do not know where they get it. ... Since we consider ourselves free and in no way bound by their injunctions, we must give other rules of definition that are consonant with true logic, in accordance with the distinction among things by their essence.” 48

Without such a distinction it is impossible to improve the intellect. After all, the intellect improves not so much by increasing the number of adequate ideas as by acquiring more concrete knowledge about something or focusing on more perfect objects.

“Ideas themselves are the more perfect, the more they express the perfection of whatever object; for we do not admire the architect who has planned a chapel so much as the architect who has planned a splendid temple.” 49 “The perfection of an idea and the effective [40] capacity for thought are assessed according to the perfection of the object.” 50 The quality of our knowledge of a thing is conditioned by the measure of perfection of the object of knowledge and, as soon as logic accepts the task of perfecting the intellect, it must in some way take into account the objective content of ideas and be able to determine the measure of perfection of the thing that is included “objectively” in the idea. 51 Spinoza’s “objective logic” is the reflective knowledge of the intellect as the idea of substance and contains principally those categories that Hegel placed in the doctrine of essence. The categories of the doctrine of being, which constitutes the first part of Hegel’s “objective logic,” Spinoza classifies partly with the reflective idea of substance understood as “an extended thing” (for Spinoza, as previously for Descartes and then for Leibniz, the nature of extension is described by the categories of quantity), partly by the abstractions of the imagination (entia imaginationis: pure being, nothing, something, and some other things), and partly by the categories of reason (entia rationis: limit, number, measure, etc.). The abstractions of the imagination and the categories of reason are examined by Spinoza in his doctrine of substance as “a thinking thing,” which he sets forth principally in the Treatise on the Improvement of the Understanding, and in Part 2 of Ethics.

In conclusion we must say something about what it is that Spinoza uses as a criterion of the “improvement” of things grasped by the intellect, since this is not only the supreme principle of Spinoza’s logic, but also an idea that has a definite role in the development of theoretical thought in general, that is, in the actual process of “improving the intellect.”

“The most perfect being” is construed in this logic as an “absolutely infinite thing” (see the definition of God in Ethics), and the perfection of any separate thing is determined by the degree to which it “expresses” (exprimit) 52 the infinite nature of God. The old postulate of the logical primacy of the idea of the absolutely infinite found its ultimate embodiment in Spinoza’s philosophy (two centuries earlier than in mathematics, which Spinoza praised as the “model of truth” (veritatis norma) for its logical merits 53). True, Spinoza was unable to explain the origin of this idea in the human mind and considered it, in line with the Platonists and Descartes, to be “innate.” Thus he seemed to admit that he does not know how the mind acquires this idea; however, he is firmly convinced that an idea directly expressing absolute [41] infinity cannot have as its cause finite human thought and especially imagination, as Gassendi, Locke, and their disciples asserted, and that the human mind is capable of comprehending the truth of things only sub specie aeternitatis, that is, only to the extent that it is guided in its acts by the idea of the absolutely infinite. Descartes’s and Spinoza’s elucidation and demonstration of the latter proposition did not simply mark the beginning of the era of reformation in logic but, as the history of natural science and mathematics has shown, became one of mankind’s most important steps in the task of improving its own intellect.


1 See M. Klain [Kline], Matematika. Utrata opredelennosti (Moscow, 1984), p. 408. “Deductive logic is a child of mathematics,” declares Morris Kline boldly (ibid., p. 30).
2 R. Dekart [Descartes], “Beseda s Burmanom,” Sochineniia (Moscow, 1989‑94), vol. 2, p. 485.
3 Ibid., pp. 483‑84. Descartes developed this thought meticulously in Rules for the Direction of the Mind. Traditional logic is everywhere called “dialectic” here (see Sochineniia, vol. 1, pp. 81, 87, 109‑10, 127]. The author in this way seems to insinuate that he considers existing theories of the method of thought to be unworthy of the name logic.
4 R. Dekart, “Pravila dlia rukovodstva uma,” Sochineniia, vol. 1, p. 88.
5 R. Dekart, “Rassuzhdenie o metode,” Sochineniia, vol. 1, p. 262.
6 Dekart, “Pravila dlia rukovodstva uma,” p. 92.
7 Ibid, p. 93.
8 Ibid.
9 Ibid, p. 118.
10 Just as definitely, Spinoza sets out his thought on the mutual transformation of the universal and the individual in theorem 28, part 1 and in theorem 9, part 2 of Ethics. He says there that in regard to singular things and their ideas, God, that is, the common cause of all that exists, appears also in the guise of some singular thing or idea, and hot in the capacity of an absolutely infinite thing, as God, according to definition 6 of part 1, is “in Himself.” God “exists as an affect” (affectus est) of an singular thing to the extent that it appears as the cause of other things. In this sense God is nothing other than the universal causal connection or the universal law of nature. It is clear that the relationship between the categories of the universal and the individual is understood here very differently than in traditional logic. To signify a true universal Spinoza uses the term communis, and to signify the nominal‑general, that is, the species‑genus abstractions of similar attributes, he uses the term universalis. He cautions more than once against the inadequacy of any inference ex abstract is, that is, on the basis of these abstractions, whereas a true universal, according to theorem 38, part 2 of Ethics, can be understood only adequately.” (For more details on this see A.D. Maidanskii, “Proiskhozdenie definitsii u Spinozy,” Vestnik MGU. Seriia 7 [42] (Filosoflia), 1995, no. 1, pp. 55‑60.
11 Dekart, “Pravila dlia rukovodstva uma,” p. 140.
12 Ibid, p. 147.
13 M. Monten’ [Montaigne], Opyty (Moscow, 1979), p. 266.
14 R. Dekart, “Razmyshleniia o pervoi filosofii,” Sochineniia, vol. 2, p. 20.
15 M. Khaidegger [Heidegger], “Evropeiskii nigilizm,” Problema cheloveka v zapadnoi filosofii (Moscow, 1988), p. 262.
16 Dekart, “Razmyshleniia,” p. 38.
17 Ibid, p. 40.
18 M. Gueroult, Descartes selon l’ordre des raisons (Paris, 1953), vol. 1, p. 155.
19 E. Curley, Descartes Against the Skeptics (Cambridge, 1978), p. 94.
20 R. Dekart, “Vozrazheniia nekotorykh uchenykh muzhei protiv izlozhennykh vyshe ‘Razmyshlenii’ s otvetami avtora,” Sochineniia, vol. 2, p. 89.
21 Dekart, “Razmyshleniia,” p. 53.
22 G.V.[W.] F. Gegel [Hegel], Nauka logiki, 3 vols. (Moscow, 1970‑72), vol. 3, p. 152.
23 F.V.I. Shelling [Schelling], Sochineniia, vols. (Moscow, 1987‑89), vol. 2, p. 398.
24 I. Kant, Sochineniia, 6 vols. (Moscow, 1964‑66), vol. 3, p. 524.
25 At the beginning of the sixties a few well‑known philosophers took part in a discussion of the a priori argument in the Philosophical Review. The occasion for this polemics was an article by Norman Malcolm (“Anselm’s Ontological Argument,” Philosophical Review, 1960, vol. 69, pp. 41‑62). Vol. 70 (1961) of the journal was devoted to this discussion.
26 Dekart, “Razmyshleniia,” p. 54.
27 R. Descartes, Oeuvres, 12 vols., eds. C. Adam and P. Tannery (Paris, 1897‑1913), vol. 7, p. 68.
28 Dekart, “Razmyshleniia,” p. 56.
29 Ibid., p. 43.
30 Ibid.
31 Ibid, p. 57.
32 See B. Spinoza, “Kratkii traktat o Boge, cheloveke i ego schast’e,” Izbrannye proizvedeniia (Moscow, 1957), vol. l, p. 106.
33 All currently existing translations of Spinoza leave something to be desired. Since this is not the place for a thorough examination of the errors and omissions committed by translators, we shall simply limit ourselves to presenting the corrected translation and in the cases where the correction is a major one a parallel reference to the original text (from the edition: Benedicti de Spinoza, Opera quotquot reperta sunt, eds. J. van Vloten and J. P.N. Land (Hagae Comitum, 1895), 3 vols.)
34 B. Spinoza, “Bogoslovsko‑politicheskii traktat,” Izbrannye proizvedeniia, vol. 2, p. 64.
35 B. Spinoza, “Tractatus de intellectus emendatione,” Opera, vol. 1, p. 32; “Traktat ob usovershenstvovanii razuma,” Izbrannye proizvedeniia, vol. 1, p. 356.
36 Ibid, p. 12 (p. 331).
37 Ibid, p. 10 (p. 329).
38 V. Klever, “Material’naia logika v filosofii Spinozy,” Istoriko‑filosofskii ezhegodnik ’88 (Moscow, 1989), p. 339. Klever says that in its main features this [43] view anticipates the logical program of the late Wittgenstein, so that the latter appears as a kind of “crypto‑spinozist,” and his On Certainty “is the best and most illustrative commentary on Spinoza’s philosophy” (W. Klever, “Axioms in Spinoza’s Science and Philosophy of Science,” Studia Spinozana II, Alling, 1986, p. 188).
39 Ibid, p. 334.
40 B. Spinoza, “Tractatus de intellectus emendatione,” p. 12; “Traktat ob usovershenstvovanii razuma,” p. 331.
41 Ibid, pp. 32 (p. 355).
42 V.N. Polovtsova draws an interesting analogy between the idea of healing and cleansing the intellect and Plotinus’s catharsis — the purification of the mind of the forms of sensory perception. Cf. V.N. Polovtsova, Foreword to B. Spinoza, Traktat ob ochishchenii intellekta (Moscow, 1914), p. 54.
43 Cf. B. Spinoza, Pis’ma, no. 37 (to I. Bouwmester).
44 Spinoza, “Traktat ob usovershenstvovanii razuma,” p. 325.
45 Ibid, p. 32, 34 (p. 356, 358).
46 Spinoza, Izbrannye proizvedeniia, vol. l, p. 168.
47 I would count Kant’s transcendental logic, Fichte’s theory of science, Hegel’s and Marx’s dialectical logic, and Husserl’s phenomenology among the later varieties of object‑sensitive logic.
48 Quoted in: V.N. Polovtsova, “K metodologii izucheniia filosofii Spinozy,” Voprosy filosofii i psikhologii, bk. 118 (1913), p. 360.
49 B. Spinoza, “Tractatus de intellectus emendatione,” p. 32; “Traktat ob usovershenstvovanii razuma,” p. 357.
50 Izbrannye proizvedeniia, vol. 1, p. 520.
51 The scholastic term “objectively” often encountered in Descartes and Spinoza means “given as the object of some idea.”
52 On the category expressio see F. Kaufmann, “Spinoza’s System As Theory of Expression,” Philosophy and Phenomenological Research, 1940, vol. 1, no. 1, p. 83‑97.
53 Alexandre Koyré wrote that since the times of Georg Cantor it has been accepted as firmly proven that “in the logical construction of arithmetic the concept of the infinite and the theory of infinite sets must be included in the theory of finite integers for, being logically prior to the latter, they serve as its foundation” (A. Koire, Ocherki istorii filosofskoi mysli, Moscow, 1985, p. 430). Thus mathematics, grasping the infinite only in its quantitative expression, confirms in its own way the certainty of the idea stated earlier by philosophers of the infinite in general, in its ideal form. [44]