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Kant shows that the one necessary, non-contingent existence is God, a being that is one, simple, unchangeable, eternal, and a spirit. There is, then, necessarily a God, a being comprehending not all, but all the highest positive reality…

immanuel-kantI hope that this consideration of a peculiar little work of great interest will appeal to readers who want a taste of Kant’s early work as well as to those who like to ruminate on the meaning of the simplest mathematical notions. My text is an essay called “An Attempt to Introduce the Concept of Negative Magnitudes into Philosophy (Weltweisheit).” Its date is 1763.

Its appeal to me has these three aspects: First there is the tentative mode expressed in the title; here we hear Kant’s pre-critical voice—not yet the magisterially conclusive notes of the Critique of Pure Reason (1781) but a tone at once spiritedly daring and gropingly uncertain. A second and more specific aspect is the inchoate appearance of major elements of the first Critique; not only can we see its elements come into being, but we can watch at their incipiency topics that Kant will return to all his life. The third appealing aspect of the essay is its pedagogic suggestiveness; for a teacher seeking to help students reflect on mathematical formalisms, it is a useful source.

The piece on negative magnitudes has been eclipsed by its much longer contemporary, “The Only Possible Basis of Proof for a Demonstration of God’s Existence (Dasein),” dated 1763. This essay was subjected to an extensive and deep analysis by Heidegger in his lecture course of 1927 (The Problems of Phenomenology, translated and edited by Albert Hofstadter, Indiana University Press, 1982, ¶8); his elucidation of Kant’s use of the term “reality” is particularly relevant to the essay on negative magnitudes.

What evidently drew Heidegger’s attention to this particular essay, however, was its brisk definition of Dasein—his central word—as “absolute position.” When the verb “is” is used not as a copula to relate a subject to its predicate as in “God is omnipotent” but is asserted abruptly, absolutely, as in “God is” or “God exists,” it signifies, Kant claims, a mere positing of an object. By this, in Heidegger’s interpretation, Kant means that the object is affirmed by a knowing subject as available to perception. (I observe, incidentally, that in order to express the character of existence as non-attributive absolute position more adequately, Kant proposes language that anticipates the existential quantifier of propositional logic: We should say not “A narwhal is an animal” but “There exists an animal, the narwhal, which has unicorn-attributes.”) In other words, existence is not a predicate and adds no objective attribute to God’s essence. Since it is the crux of what Kant first called the “ontological argument” (whose best known proponent is Anselm), that existence is a necessary attribute of God’s essence, Kant’s understanding of existence as a non-predicate seems to be a rejection of that proof.

Kant’s own demonstration calls on the concept of a “real-ground” (Realgrund), a concept that emerges in the essay on negative magnitudes, to be discussed below. This concept in turn involves the postulate that essence is prior to existence and actuality to possibility. Since Heidegger’s thinking is dominated by the reverse claim, he is a severe if respectful critic of Kant’s understanding. He regards Kant’s exposition of existence as a half-way house, situated between the notion of existence as one predicate among others and his, Heidegger’s, own understanding of Dasein as “extantness,” i.e., “being-at-hand,” with respect to things and “being in the world” with respect to human beings.

There is yet another contemporary essay that has bearing on the essay about negative magnitudes, the “Enquiry Concerning the Evidence of the Principles of Natural Theology and Morality, in Answer to a Question Posed by the Royal Academy of the Sciences at Berlin for 1763.” The aim of this essay (which did not win the prize) was to establish what evidence and certainty natural theology is capable of; it is thus a discourse on method. It begins with an investigation of the difference between mathematics and metaphysics (a difference that plays a major role especially in Kant’s last work, the Opus Postumum).

Thus, both pieces, the one on God’s existence and the other on theological certainty, illuminate the essay on negative magnitudes, the former through its concept of an ultimate reality and the latter through its restrictions on the use of mathematics in first philosophy. I will draw on them in my exploration.


Anyone who has stepped out for a moment from the routine familiarity of operations with signed numbers will have wondered just how, say, 5, +5, |5| and -5 differ from each other, and, furthermore, whether +5 and -5 are operations on or qualifications of the number 5. Although his essay concerns numerable magnitudes, especially those discovered in nature, questions of that sort seem to have been going through Kant’s mind, as he considered the illuminations that the actual quantification of experience might offer to philosophy.

In the essay on natural theology, Kant sets out four definitive reasons why the mathematical method is inapplicable to philosophy and is not the way to certainty in metaphysics (¶1-4):

1.) Mathematical definitions are “synthetic,” in the sense that the mathematician does not analyze a given concept, but first synthesizes or constructs it, i.e., puts it together at will. (In the first Critique synthesis will have acquired a deeper meaning; it will no longer mean arbitrary construction but an act of the understanding expressing in the imagination the formative givens of the intuition.) In philosophy or, as Kant says interchangeably, Weltweisheit, “world-wisdom” (as distinguished from the scholastic philosophy of mere, unapplied concepts, see Logic, Intro. 3), on the other hand, definitions are analytic, in the sense that concepts are given to, not made by, the philosopher, and he then endeavors to analyze them into their implicit elements. (In the Critique a way will be found for the philosopher too to form pure synthetic judgments.)

2.) Mathematics is always concrete, in that the arithmetician symbolizes his numbers and operations perceptibly, and the geometer visibly draws his figures. (In the Critique, these inscriptions will be within the field of the imaginative intuition.) Philosophers, on the other hand, use words exclusively, and these signify, in Kant’s understanding, abstractly, non-pictorially. (In the Critique this rift between word and picture is closed.)

3.) The mathematician tries to employ a minimum of unproved propositions (i.e., axioms and postulates), while the philosopher makes indefinitely many assumptions, as needed. (In the Critique the principles of experience will be systematically restricted.)

4.) The objects of mathematics are easy and simple (!), those of philosophy difficult and involved.

In the Preface of “The Attempt to Introduce the Concept of Negative Magnitudes into Philosophy,” Kant accordingly eschews the introduction of mathematical method into philosophy, but censures the neglect of the application of mathematical matter to the objects of philosophy. The really useful mathematical doctrines are, however, only those that are applicable to natural science. (This restriction prefigures the use of mathematics in the Critique, where its role is the constitution and understanding of the system of nature, i.e., of matter in quantitative and qualitative change.) As a preliminary example, Kant gives the continuity of space, which is, if inexplicably, postulated in the Euclidean geometry he assumes. The concept of continuity will give insight, he thinks (but explains no further), into the ultimate ground of the possibility of space. (The claim does prefigure the arguments for the establishment of a spatial intuition in the Critique.) The main example in this piece will, of course, be the concept of negative magnitudes, which Kant now proceeds to clarify and apply. I shall follow his arguments through the three sections of the essay.

First Section

There are two types of opposition: logical, through contradiction and real, without contradiction. If contradictories are logically connected, the result is a “negative nothing, an unthinkable” (nihil negativum irrepresentabile; now as later in the Critique, representabile = cogitabile, i.e., to think is to represent in the cognitive faculty). Thus, a body in motion is a “something” (which is in the Critique the highest objective concept, i.e., that of an object in general); so is a body at rest. But a body at the same time in motion and at rest is an unthinkable nothing. It is not so much a non-object incapable of being as a less-than-nothing incapable of being thought.

Real-opposition (Realentgegensetzung), on the other hand, involves no contradiction and thus no unthinkability. For this opposition does not cancel the being of the object thus qualified; it remains a something and thinkable (cogitabile). Suppose a body impelled in one direction and also driven by a counterforce in the other. The resulting motion may be none = 0, but the body so affected is not a nothing. Kant calls this result nihil privativum representabile, where privativum has a dynamic sense, as of an achieved condition. This nothing is to be termed zero = 0. (The distinction of “logical” and “real” underlies the Critical difference between merely analytic and synthetic, i.e., ampliative, judgments.)

Kant also refers to real-opposition as real-repugnance (Realrepugnanz) because two precisely antagonistic predicates of an object cancel each other, though they do not annihilate the object they qualify. Moreover, it is somewhat arbitrary (or rather, determined by extraneous human interests) which pole is called negative. For example,“dark” may seem to us intrinsically negative, but it is cancelled by its own negation “not-dark” (which might, of course, actually be light). Thus, one may say that in real-repugnance both opposing predicates are affirmative.

Kant gives, among others, the following example: Suppose a person owes 100 dollars and is, at the same time, owed this sum. This debtor-lender is worth zero dollars, since the two conditions cancel each other, but this fact does not cancel him, the bearer of these modifications.

Kant then offers an example (slightly adjusted by me) that expands the concept of real-opposition. Suppose a vessel going from Portugal to Brazil is carried due west by an east wind with an impelling force that would cause it to run twelve miles on a certain day, and is also subject to a countervailing current retarding it by five miles; the boat’s total progress is seven miles a day. Here the result is not = 0. It is clear that Kant regards real-opposition as taking place along a scaled spectrum of quantifiable qualities at whose center there might be (though there is not always) a neutral fulcrum, 0.

If there is an “origin,” then on one side there are the positive quantities, on the other those that can be regarded either as the relative negatives of the former, or as opposed positives in their own right. Thus, the opening question, what is really meant by +5, -5 and |5|, might be answered by Kant like this: The plus or minus sign is neither an implicit operation nor a qualification of the number as itself inherently positive or negative, for the number is, like its “absolute” expression |5|, always positive. The plus and minus signs signify rather the relation of numbers to each other: -5 is the negative of +5; it is negative only in relation to a positive five. To be sure, since Kant is not speaking of bare numbers but of magnitudes symbolizing quantified properties such as are representable along one dimension, magnitudes in real-opposition, the application to pure numbers is conjectural.

Objects in real-opposition have, of course, many negations besides those directly opposing a positive: a ship sailing westward is also not sailing southward, but its course is in real-opposition only to the eastern direction. Moreover, there are cases, say of lack of motion, which are not the result of real opposing forces but of a total absence of impelling force. Negation that is the result of real-opposition Kant calls privatio, that which has no positive ground is called defectus or absentia. His example of an absent or null result is what we would call potential energy (not to be confused with “potential-opposition,” see below): “Thus the thunder that art discovered for the sake of destruction lies stored up for a future war in the threatening silence of an arsenal of a prince, until, when a treacherous tinder touches it, it blows up like lightning and devastates everything around it.” Aside from Kant’s pacifistic poetry, this example shows that Kant is considering only the magnitudes of actualized forces.

It is pretty clear that what Kant is struggling to do is to present a conceptual underpinning for what we call directed magnitudes as they occur in the world, including signed numbers insofar as they represent natural qualities, whose plus or minus tells us whether we are to move respectively to the right or the left along the line-spectrum (conceived as a straight line, where left is negative by convention). Not that Kant is thinking of our mathematical number line—he does not even mention an origin (=0), and his opposition-spectra are evidently not necessarily infinite in either direction. While he does insist on the relative directionality of negative numbers, insofar as they countermand their positives, he also reiterates that the negatively directed magnitudes are not negative numbers insofar as these are regarded as being less than 0.

For the use of directed magnitudes in philosophy it will, in fact, be essential that the opposed quantities are indeed inherently positive, as will be shown in Section Three. Hence a debt can be called negative capital, falling negative rising, and so on, where it is our perspective that gives a negative emotional tint to one of the terms. Kant sums his view up in a basic law and its converse: 1. Real-repugnance takes place between two positives, power against power, which cannot be contradictories but must be of the same kind (while the complement class in a contradiction, e.g., “non-dark objects,” is not necessarily of the “same kind” as “dark objects”; they might be invisible objects). 2. Where a positive opposes its proper positive, a real cancellation will occur.

Furthermore, certain rules of operation follow. For example, if the opposites are quantitatively equal, their sum will = 0, or A – A = 0, which shows, Kant explains, that both A’s are positive (since A = A). Also A + 0 = A and A – 0 = A, since no oppositions are involved. But 0 – A is philosophically impossible since positives cannot be subtracted from nothing: There are no inherently negative qualities and so, as was said, no directed magnitudes inherently less than 0 (!). This odd-sounding but unavoidable consequence will have important metaphysical implications.

Real-repugnance, strange though its label be, has an Aristotelian antecedent. It is, I want to argue, a dynamic version of the logical opposition Aristotle calls contrariety (Metaphysics 10.4). Kant himself says as much in his Anthropology of 1798; he there contrasts contradiction or logical opposition (Gegenteil) with contrariety or real-opposition (Widerspiel, ¶60).

As a formal logical opposition contrariety occurs in the Square of Opposition, which is a tabulation of the Aristotelian doctrine on the subject: If the basic proposition is “Every S is P,” its contradictory is “Some S is not P” and its contrary is “No S is P.” Contradictories cannot both be true nor can they both be false, but contraries, though they cannot both be true, can both be false; for if not every S be P, yet might it be false that no S is P. Contrary propositions bear a certain formal relation to contrary terms, for these cannot both at once belong to an object but they might both fail to belong to it. Thus, an object could not be at once pitch black and pure white, but it need not be either, which is untrue of contradictories, such as black and not-black.

The above paragraph is really a digression to show that it is not contrary propositions but contrary terms denoting qualities that are related to Kant’s real-opposition. Contraries are qualities that are not simply in abrupt polar opposition (though their extremes delimit a maximum difference) but are connected through a spectrum of gradations. Aristotle makes privation a particular case of contrariety, and for Kant negation, interpreted as privatio, in German Beraubung, meaning “deprivation,” introduces into some of the ranges of opposition a kind of null point or zero through which the quality goes by degrees into a negative or oppositional mode. (As was noted, not all the spectra have such a center; for example when bodies are brought to rest = 0 by countervailing forces this 0 is not an origin in the spectrum of forces but is a net effect in the bodies’ position, measured in spatial extension rather than as qualitative intensity.) Some of Kant’s examples will be given below.

What turns contrariety into real-repugnance is the dynamic view Kant takes of this opposition: It means not just being supinely, matter-of-factly opposed; it means being antagonistically, aggressively opposed. Moreover, this striving in many different dimensions of quality is quantifiable in degrees of intensity, as Kant’s examples will show.

Second Section

Kant now calls for examples from 1. physics, 2. psychology, 3. morality.

1.) In physics his prime example is impenetrability, which is a positive force, a true repulsion that might thus also be called “negative attraction.” For attraction is a cause, contrarily directed, by which a body compels others to push into its own space.

2.) From psychology (Seelenlehre) come Kant’s most pungent examples of real-opposition. He raises the question whether aversion could be called “negative pleasure.” The fact that in German aversion, Unlust, looks like a direct contradictory of pleasure, Lust, gives him pause, but he observes that in “real-understanding” (Realverstand), meaning in actual psychic perception, aversion is not just a negation of desire or even its diminution, but a real-repugnance, a positive perception. Then, to illustrate, comes an almost comical quantification: A Spartan mother hears of her son’s heroism; a high degree of pleasure ensues, say of 4 degrees. Then comes the news of his death. If the resulting Unlust were a mere negation, a mere negation of the Lust, it would equal 0. But 4 + 0 = 4, as if the death made no difference to her delight. Kant concedes however (some notion of a mother’s feeling!) that the positive Unlust of his death, the real-repugnance, will diminish the mother’s Lust at her son’s bravery by one degree; it will, therefore, = 3.

In the same vein, disgust is negative desire, hate negative love, ugliness negative beauty, error negative truth, and so on. Kant warns against regarding this terminology as mere word mongering: It is a philosophical pitfall to regard the evils of positive privation as mere defects. Thus Kant is denying, surely quite incidentally, the theological doctrine (found in Plotinus, Augustine, Thomas) of evil as privation of good (privatio boni), for, in Kant’s view, in this essay evil, though a relative negation, is yet a positive force.

3.) Kant naturally regards his concept of real-opposition as having important uses in philosophy insofar as it is “practical prudence” (Weltweisheit), that is, applied morality. Non-virtue (Untugend) is, in human beings, not a mere denial of virtue (Tugend), but a positively negative virtue, vice. This is the case because humans, unlike animals that are morally unendowed, have an “inner moral feeling” that drives them to good actions. For instance they harbor a law of neighborly love. To do bad deeds, human beings have to overcome this natural inclination to good. (A quarter century later, in the second Critique, the Critique of Practical Reason, 1788, Kant will, on the contrary, see the test of true morality in the overcoming of natural inclinations.)

Thus, certain people must make a noticeable effort to engage even in sins of omission (such as neglecting to offer neighborly help), which differ from sins of commission (hurting one’s neighbor) only in degree; hence, the slide from the one to the other is all too smooth, though the beginning is effortful.

Kant apologizes for what may seem to enlightened readers the prolixity of his exposition. He is writing for a “indocile breed of judges, which, because they spend their lives with a single book, understand nothing but what is contained therein.” The book is, I imagine, the Bible.

In an appended remark, Kant forestalls the notion that the world conceived in such a dynamic balance is capable neither of augmentation nor perfection. He points out (1) that in potential oppositions (to be explained below) though the total quantity of effect may = 0, yet there may be an increase in apparent change, as when bodies widen the distance between them; (2) that it is the very antagonism of natural forces that keeps the world in its perfectly regular courses; and (3) that though desire and aversion do balance each other considered as positive quantities oppositely signed, who would claim that aversion is to be called a perfection? Moreover, though the net quantity of moral action in two people may be the same, yet the quality of the one who acted from the better intention is to be more greatly valued. Kant adds that these calculi do not apply to the godhead, which is blessed not through an external good but through itself.

Third Section

Kant introduces this section, which contains his startling application of the concept of real-opposition to metaphysics, by insisting once more that it is a mere attempt, very imperfect though promising: It is better to put before the public uncertain essays than dogmatically decked-out pretenses of profundity. This section is accordingly called “Containing Some Reflections Which Can Be Preparatory for the Application of the Concept Here Thought Out to the Objects of Philosophy.” I dwell on this language because it is not a tone familiar to those of us who have spent time with the three Critiques.

1.) Everyone easily understands how it is that something is not—the positive ground for its being is absent; there is no reason for it to be. But how does something cease to be? The question arises because we must understand every passing-away as a negative becoming. As such it requires a real or positive ground.

In the first two sections, Kant had spoken of real-opposition or real-repugnance, by which were meant normally apprehensible contrarieties. Now, in the third section, he introduces real-grounds (Realgründe). As far as I can make out from the examples, real-grounds are natural, including psychic, causes: forces, powers, and acts exercised by material or psychic agents. “Real” here is used not as in real-opposition, which is opposed to logical contradiction, but it seems to mean “affecting perceptible existence.” Real-grounds seem to be a first-level, underlying causal reality, apprehended through its effects. Thus Kant uses “reality”—and, as we shall see, “existence,” “being” (Wesen), as well as “actuality” (Wirklichkeit)—quite loosely in these exploratory essays; Heidegger shows that Kant’s later systematic meaning of reality is the “whatness” of a thing, all its possible predicates, its essence, while existence is perceptibility.

Kant’s examples are mainly from the soul. It costs real effort to refrain from laughing, to dissipate grief, even to abstract from a manifold representation for the sake of clarity; thus abstraction is negative attention. Even the apparently random succession of thoughts has real grounds, which are “hidden in the depths of the mind (Geist),” i.e., in what we call the subconscious. Whether the change is in the condition of matter and thus through external causes, or of the mind and thus through inner causes, the necessity for a causal real-opposition remains the same.

Kant is focusing here, it seems to me, on a partial converse to, and a kind of complement of, a question, evidently not asked by the ancients, which is to become a modern pre-occupation: “Why does something exist rather than nothing?” It was first raised as a metaphysical problem with a theological answer by Leibniz (The Principles of Nature and Grace, Founded on Reason, 1714), and was repeatedly taken up by Heidegger (especially in the end of What is Metaphysics, 1949), who treated it as “the basic question of metaphysics,” though to be resolved without recourse to theology.

Leibniz asks for a reason, sufficient and ultimate, to account for the existence of a universe of things in progress and finds it in God. Kant, on the other hand, asks for an adequate reason why a present condition in the world should go out of existence and finds real-opposition as the cause. It is, however, an unexplained cause, which, Kant says at the end of the essay, he has been and will be thinking about and will in the future write about. This appears to be a harbinger of Critical works not to appear for almost two decades. Meanwhile the metaphysical consequences are exceedingly strange: The sum total of all real grounds equals zero, and “the whole of the world is in itself Nothing.” More of this below.

2.) Kant says that the theses to be here proposed seem to him “of the most extreme importance.” First, however, he distinguishes real-opposition, or as he now calls it, “actual” opposition, from “possible” or “potential” opposition. The latter type of opposites are also each other’s negatives, real to be sure, but not, as it happens, in conflict. Thus two forces, each other’s opposites, may be driving two different bodies in opposite directions: They have the potential to cancel each other’s motion but do not actually do so in the given situation. So also one person’s desire may be the other’s aversion, yet their ability to stymie each other is only a possibility. Kant then offers the following first general thesis: “In all natural changes of the world their positive sum, insofar as it is estimated by the addition of agreeing (not opposed) positions and the subtraction from one another of those in real-opposition, is neither increased nor diminished.”

Recall Newton’s Third Law of Motion (Philophiae Naturalis Principia Mathematica, 1687): “To every action there is opposed an equal reaction: or, the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.”

It is pretty evident that Kant is struggling to ground in metaphysics and apply to the human world the most dynamic of Newton’s Laws of Motion or perhaps the Law of the Conservation of Momentum, which is implied in (or, some argue, implies) the law of equal action and reaction; it states that when the forces acting on bodies in the same and in contrary directions are summed (over time) the quantity of motion (i.e., the mass compounded with velocity = mv) is not changed. Kant says, without proof, that although this rule of mechanics is not usually deduced from the metaphysical ground from which his first thesis is derived, yet it could be. (The first Critique will be partly devoted to giving the transcendental grounds of the laws of motion and conservation.) Not only bodies in motion but souls in emotion obey the law of conservation, and so do humans in action. Astoundingly, Kant really means this: For every “world- change,” i.e., every “natural” change (which includes the psychic realm) there is an equal and opposed change, so that the sum of measured final positions, i.e., states of existence taken globally, is equal to what it was before the change, or the total effect = 0.

Every becoming, then, induces an actual or potential counter-becoming or cessation. It can now be seen why Kant introduced potential real-opposition: Kant’s forces not only act in opposed pairs but they may act on different objects, i.e., the real-opposition may not be actualized in one body or one soul or even in the same mode in one soul. Nevertheless, these potential oppositions enter into the summed effects of the world-total.

Kant then goes on to give more concrete non-mechanical examples—though these are not non-natural, because (as will still be true in the Critiques) the soul, excepting in its practical-moral employment of reason, is subject to natural dynamic forces. The examples of this third section differ from the ones in the previous section exactly as the course of the exposition requires: They are cases not of actual but of potential real-opposition. Thus, if one person’s pleasure and displeasure arise not from the same object, but the same ground that caused pleasure in one object is also the true ground for feeling displeasure in another, then in analogy to two bodies moving in contrary directions by repulsion, there is one potential real-ground and cause for the positive and the negative feeling. These feelings oppose but also bypass each other; they may cancel each other but need not do so. This is why the Stoic sage had to eradicate all pleasurable drives—because they always engender an associated but diverse displeasure that affects the final value of the pleasure, though perhaps not the actual pleasure itself. Even in the use of the understanding, we find that to the degree that one idea is clarified, the others may be obscured, though surely not by the clarification. Kant adds that in the most perfect Being the zero sum result does not hold, as will be shown.

Now comes the second thesis, which is simply the translation of the first thesis into its causative grounds: “All real-grounds of the universe, if one sums those similarly directed and subtracts those opposed to one another, yield a result which is equal to zero.”

Kant immediately states: “The whole of the world is in itself nothing, except insofar as it is something through the will of someone other.” Regarded by itself, the sum of all existing reality = 0; the world is an almost Heraclitean system of balanced oppositions, of mutually negating positivities. In relation to the divine will the sum of all possible reality, of the world’s existence, is, however, positive. But it itself is not therefore in real-opposition to the divine will; it is not the godhead’s relative negation. Consequently existence, i.e., whatever is perceptibly there in the world, is through its internal relations nothing, but in relation to the grounding will of the divinity it is something; it is positive. For there can be no real-opposition of the world to the divine will. (This thought is still to be found in the Opus Postumum; there too the real-opposition of forces is metaphysically exploited.)

The nullity of the physical and psychical world in its summed effects, the nothingness of the underlying universe of summed causes, and the positivity of creation only in relation to God—Kant presents these results without any discernible pathos, without acknowledgment that this cancellation of the world-whole of effect and cause might bear a religious or moral interpretation beyond the intellectual proof of God’s existence. Nor can I discover that he ever reverted to this nullifying construal of the laws of conservation. Perhaps it is to be regarded as a passing notion that served as a spur to further inquiry into the world’s relation to its ground. (Kant does hold on to the “law of the antagonism in all community of matter by means of motion”; any divergence from its reciprocity would, he now argues, move the very center of gravity of the universe, Metaphysical Foundations of Natural Science of 1786, ¶563. In this essay too the notion of real-opposition is put to work in the specific antagonism of repulsion and attraction, forces which between them are responsible for the way matter, i.e., “the movable,” fills space. However, the explanation of these forces in their specificity remains an unsolved problem for Kant into the Opus Postumum.)

So Kant concludes by making explicit the heretofore tacitly assumed converse of the proposition that the real-grounds are responsible for the nullity of existence: Because the internal sum of existence is zero, it follows that the “real-grounds,” i.e., the forces and powers producing effects in the world, must be in a corresponding opposition: The realm of existing and possible reality is in itself shot through with grounding polarities that cancel existence, though, with respect to the divine will (Kant does not speak of “God” in this context), it has positive being (Wesen). This overt conclusion of the last section is surprising since it follows close on the assertion that the zero sum of all existence flows necessarily from the grounding being (Wesen) of the world; it is hard to tell whether we are to infer from the zero sum of effects to the underlying real-opposition of causes or whether these causes are posited first. I hesitate to detect Kant in unwittingly circular reasoning here. (He is, to be sure, the master of intentional circularity in the Critique, where the grounds of the possibility of experience are inferred from experience while experience is certified by the grounds. Perhaps the apparent circle in the above paragraph is a precursor of critical thinking.)

The metaphysical intention is however quite clear: (1) The world exists as a complex of quantifiably opposed effects; negative magnitudes express such relative opposition; when summed with their positives they yield zero. (2) Underlying these existences, there is a realm of grounds; these are forces, powers, and actions; they are also in mutually cancelling opposition, and, like their effects, they are so only relative to each other. (3) With respect to an ultimate ground, they are positive, but since no real-opposition to it is possible their positivity is not a relation of opposition to the divinity. Kant himself knows that he has not yet sufficiently clarified the character of real-grounds, nor their relation to the divine will.

His first definition of a real-ground actually occurs in a General Remark appended to the third, final section of the essay. A logical ground is one whose consequence can be clearly seen through the law of identity; for example composition is a ground of divisibility, i.e., it is identical with part of the meaning of the concept and can be educed from it analytically. A real-ground, by contrast, has a relation to its effect which, although quite truly expressed as a concept, yet allows no judgment, no true understanding, of the real-ground’s mode of action. In other words, the relation of a real cause to its effect is not apprehensible by mere logical analysis. (Here, of course, is formulated the problem that Kant will solve in the Critique by means of the “synthetic judgment a priori”—the cognition in which real connections are made—not, of course, by the logical law of identity but through our cognitive constitution.)

In this essay, this understanding of a real-ground raises—for the first time for Kant—the fundamental question of causation: “How should I understand it THAT, BECAUSE THERE IS SOMETHING, THERE IS SOMETHING ELSE?”

The will of God is something. The existing world is something else altogether. Yet, the will of God is the ultimate real-ground of the world’s existence. Kant says that no talk of cause, effect, power, and act will help: God and world are totally each other’s other and yet through one of them the other is posited. And the same holds on a lower level for the natural causes in the world, be they of an event or its cancellation. Kant promises an explication in the future, but, for now, he remarks only that the relation of real-ground to what is posited or cancelled through it cannot be expressed in a judgment, i.e., a mental “representation of the unity of a consciousness of different representations,” but is in fact only a mere, non-analyzable concept, that is, a general representation or a thought (Logic ¶1, 17). Kant is saying that the relation of an effect to its cause cannot be articulated as an affirmed attribution of a predicate to a subject. It is his way of expressing Hume’s rejection of empirically grounded causation. (In the Critique the attempt to make God’s causal relation to the world comprehensible to reason will be shown to be hopeless, but causality within the world will be grounded in the very constitution of the spatial  intuition.)

So ends the essay in which the reflection on negative magnitudes has led, through the concept of real-opposition, directly to the problem of causal connection and indirectly to God as ultimate cause. I have not done justice to the exploratory tone of the essay, to Kant’s witty derision of those who get stuck in premature dogmatism, and to his sense of having made a mere, even insufficiently explicated, beginning, but a beginning of something very important: the inquiry into causation.


An elucidation of some of the matters left unclear in the essay on negative magnitudes occurs in the essay on “The Only Possible Grounds of Proof for a Demonstration of God’s Existence” of the same year (1763), and it seems to me so daring that I cannot resist carrying this exposition a little further. The reflections I shall refer to are not those that most interested Heidegger, the ones concerning existence as position, but those dwelling on the relation of possibility to actuality (or as Kant says, to existence) and on an absolutely necessary “existent” (Dasein; First Part, Second and Third Reflections).

Possibility, Kant says, depends entirely on the law of contradiction: That is a possible something the thought of which accords with what is thought in it. This “comparison” of a subject with its predicates through the law of contradiction is to be called logical or formal possibility; a triangle cannot have other than three angles, for that would contradict its definition. But it also has something additional, a given character as a triangle in general or in particular, say a right-angled triangle; these are the data of its material or real possibility.

Therewith possibility without prior existence is abolished. For a thing is not impossible—a “no-thing”—only when contradictory predications are made of it, when it is formally impossible, but also when it offers no real material, no data, to thinking. For then all thinking ceases, for everything possible must be something thinkable, must offer stuff for thought. It follows that for Kant, possibility is conditioned on actuality.

There is, to be sure, no formal inner contradiction in the brute negation of all existence, since nothing has been posited to begin with. But that there be a possibility and that nonetheless nothing actual exist—that is contradictory. For if nothing exists, nothing material is given to be thought about. Therefore to say that nothing exists is, according to the previous analysis of existence (Dasein) as a positing act of thought, to think and say that there is absolutely nothing. And then to add that something is possible is clearly self-contradictory, for no material for thinking at all is given. Thinking involves material givens; without them it contradicts its own character. Thus, the cancellation of the material data of possibility also cancels possibility. It is absolutely impossible that nothing should exist. I understand Kant to mean that one can—logically—deny all existence, but having denied it one cannot then retrieve its possibility.

The only really elucidating example of a necessary existence is, Kant says, that of the unique Subject (i.e., God), to be touched on below. Meanwhile, if we ask, for example, how existence precedes possibility in respect to “body,” we may grant that the concept body contains no logical impossibility, yet to call on its predicates of extension, impenetrability, force, to be the data of possibility (either assumed or experienced) in the absence of actually existing, given bodies, is quite unwarranted. Without such data the concept “body” is empty. (We see here the forerunner of the dictum in the first Critique that the mere functions of the understanding are empty without the givens of intuition.)

Then Kant explains the concept of an absolutely necessary existence. To say that it is that whose contradictory is in itself impossible is a merely nominal explanation. Since existence is no predicate, its denial can never conflict with other predicates. However, to deny the positing of the thing itself is not a denial of predicates but of something else, and hence is not contradictory. Kant is looking not for logical but “real-necessity” (Realnotwendigkeit), for what cannot be denied in any “real-explanation” (Realerklärung). This is it: “What I am to regard as absolutely nothing and impossible must be that which eradicates all thinking.” Now total nonexistence in fact cancels all the material and data of thought, and hence it is impossible.

It follows that there is an absolutely necessary being. For all possibility assumes something actual, whose cancellation would itself cancel all inner possibility, i.e., the real coherence of predicates. That part of existence, on the other hand, which does not provide the material for all that is thinkable, but without which there would still be matter for thought—and thus possibility—that part is, although in a real sense possible, yet in the same sense conditionally possible, i.e., contingent; not all existence is necessary. (In his Inaugural Dissertation of 1770 Kant will say that all worldly substances are in fact contingent since they maintain reciprocal relations, while necessary beings are independent, ¶19; an earlier version of this existence proof is to be found in Kant’s Nova dilucidatio of 1755.)

Kant now goes on to show that the one necessary, non-contingent existence is God, a being that is one, simple, unchangeable, eternal, and a spirit. But its philosophically most important attribute is that it is (in scholastic terms) the ens realissimum, the most real being, that which contains the highest reality. For it contains all the givens, the data, of possibility either as directly determining other existences or as being the real-ground of which they are the consequences. (In the Critique, the ens realissimum will be relegated to the status of an ideal of reason, a regulative idea that marshals our thoughts of the world; in the Opus Postumum God is once more the most real existence, though one reason necessarily posits for itself.)

Does the attribution to God of the most and the highest reality mean that all realities, i.e., all real attributes, must be assigned to God? Here the concepts of the essay on negative magnitudes come into their own: It is common doctrine that one reality can never contradict another reality, since both are truly affirmed. But this assertion leaves out of account the notion of real-repugnance, i.e., of real-opposition. Realities may, indeed must, oppose each other without one of them being in itself negative. That was the essay’s main finding. In God, however, even real-opposition cannot take place, because that would result in privation or defect and would contradict God’s maximal reality. Thus God contains no realities in opposition to his positive predicates; for example, the real-oppositions attributable to bodies, such as being subject to contrary forces, cannot without contradiction belong to a being that has intellect and will; hence, the ens realissimus has clear positive determinations.

It is, it seems to me, implicit in these pre-Critical essays that neither existence (Dasein) nor actuality (Wirklichkeit) is as yet convertible with subjective perceptibility, though both will indeed be so later, as Heidegger observes. Instead, these terms mean objective givenness, thereness, be it sensorily received or essentially apprehended. The road to the first Critique will be the development of this conversion from the object of experience as given to the subject to its being constituted in the subject. The roots, however, of the primacy of the subject are already present in one respect, now to be shown.


There is, then, necessarily a God, a being comprehending not all, but all the highest positive reality. He is a real-ground of the world; the world, in turn, amounts quantitatively to a self-cancelled nothing, though it may well be qualitatively positive. One way to get hold of this—by Kant’s own frequent confession—still inchoate complex is to ask just how daring a departure from tradition it is.

Kant, who without naming Anselm is attempting to rebut his argument for the existence of God, calls his own proof “ontological” (later in the Critique that is what he will call Anselm-type proofs). Anselm argues (Proslogium 2, 4) that God is a maximal being whose essence is to be thought as largely and inclusively as possible; thus, it must include the predicate existence. This—that existence is a predicate—is what Kant denies, but he accepts something that seems to me even deeper in, or rather behind, Anselm’s argument: that when I must think that God exists, he exists. But this is thinking of the type Kant himself engages in when he makes God’s existence follow from the existential necessities of thinking: What is required for thinking to be possible must necessarily exist. (This type of proof becomes explicit in the Opus Postumum.) Here Anselm and Kant are brothers under the skin. I can think of counterarguments to their assumption (though without being quite persuaded by them): Is it utterly impossible that a being that must exist in thought fails to exist in fact—is it so totally unthinkable? Is it not possible to think that thinking can do utterly without the material, the data grounded either in a highest reality, as in the essay on God’s existence, or in some sensory influx from a transcendent outside, as in the first Critique? Is it unthinkable that possibilities do not disappear when actualities fail, but that there is spontaneous, autonomous, self-generated, worldless thinking? Kant has, it seems, levered the Cartesian-type certification for personal existence; Cogito, ergo sum, “I think, therefore I am,” into a proof of God’s existence: Cogito ergo Deus est, “I think therefore there is a God.” But what if “I think” entails instead: “I make the world,” if I myself give myself the data?

Such misgivings and intimations aside (they will become Kant’s own in the Opus Postumum), he has found an approach to a question that seems never to have occupied the ancients and was, as was mentioned, first formulated by Leibniz (Principles of Nature and Grace, 7): “Why does something exist rather than nothing?, especially since ‘nothing’ is simpler and easier than ‘something.’” Leibniz finds the answer in the ultimate sufficient reason called God. Kant argues the other way around: Nothing is harder than something, indeed impossible for thought, and God becomes necessary not as a sufficient reason inferred from the world’s existence but as a necessary being implied by human thinking. In 1763, having proved in one essay the necessary existence of the highest and most real ground and so (for the time being) answered Leibniz’s question, Kant is left with the unanswered next question of the essay on negative magnitudes quoted above, which he prints in block letters: “How can I understand THAT BECAUSE SOMETHING IS, SOMETHING ELSE MIGHT BE?” In other words, having proved God’s existence, how can I understand him, or his agents in the world, as causal grounds? (This very same question will, as was said, be presented in the Critique as unanswerable by logical thinking alone, but solved with the aid of the a priori relations given in the intuition.)

By 1781, the year of the first edition of the first Critique, Kant will have given up not only his own so circumstantially prepared ontological proof, the only possible one, as he had once thought, but also in principle any expectation of a theoretical demonstration of God’s existence—and so, it seems, any rational explanation of the first question, why there is existence at all.

The second question, on the other hand, is just what the Critique addresses. Kant distinguishes cosmological freedom, the power to make an absolute beginning, from causality according to natural law, which is rule-governed consequence acting within what already exists. An insight into the first, into absolute causation, i.e., creation, Kant shows, is in principle impossible for us, for it is beyond the limits of human reason. The second causality, that of lawful succession, of cause and effect in natural events, is grounded in the synthesizing character of our cognitive constitution. The essays here considered show Kant—and for my part I find this intellectually moving—casting about for disparate clues to the concepts and claims which would one day come to cohere in his master edifice.

Books mentioned in this essay may be found in The Imaginative Conservative Bookstore. Republished with gracious permission from the author (St. John’s ReviewVolume 48, No. 2, 2005).

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3 replies to this post
  1. 0-A is both mathematicaly and philosophicaly possible. To be honest, it seems to me that everything that is mathematically possible is by definition philosophically possible and everything which is not mathematically possible is not philosophically possible. Does this mean that since God is not mathematically possible then He is not philosophically possible and therefore impossible? No – God is not a possibility at all; He is an Actuality, not a possibility – and makes himself known to us neither through Math nor through Philosophy – where we might sometimes catch a glimpse of Him – but never know Him. God makes Himself known to us through miracles.

    But stepping away from miracles and returning to Math:

    Mathematically 0-A=-A for the same reason that 0+5 is +5 or just 5. -A is not a Natural number, but a negation of a Natural number or simply a negative Real number. This assumes A is any Real or Natural Number and that 0 is not a Natural Number (the traditional asusmption). This traditional mathematical assumption is not altogether mathematically sound and the source of debate amongst mathematicians.

    Philosophicaly it helps to understand the debate amongst mathematicians as to whether or not 0 is a Natural number. To understand this debate it is necessary to understand the basics of Set Theory. 0, if we assume it to be an Empy Set, can arguably be called a Natural number. If so, then 0-A could just as well be 0 if A=0 (if A is any Real or Natural number and 0 is a Natural number in accordance with the notion of an Empty Set). This would actually mean that 0-A has two possible answers depending on how we define A and how we define 0.

    If A is any Real or Natural Number and 0 is a Natural number (Empty Set) in accordance with Set Theory then:

    0-A= -A (Real Number)


    0-A=0 (Natural Number/Empty Set)

    Kant could not know this because he was writing before Georg Cantor and Polish mathematician Waclaw Sierpinski. Both of these gentlemen are the mathematicians chiefly responsible for set theory. Reading Sierpinski’s Set Theory and Topology is far more interesting than reading Kant.

    In general, there is an odd deficiency in Western enlightenment philosophy to look for God where He is least likely to be found – Math and Science. Math and Science, like all things of this world, reflect God to some extent – but He cannot be found in them per say, only in contemplation of the miracle of the Cross and the grace of other miracles. God does not need to be “philosophically possible” and mathematics is a gift from God too beautiful to corrupt by using it to grasp God. Kant is – as Nietzsche rightly argued – a nihilist. The entire endeavor made in the West to rationalize and mathematize God has resulted in pushing us away from God – and in ruining the level of mathematical education in the West as well – as nowadays so many religious people in the West feel the need to rebel against Math and Science in favor of God.

    The Eastern approach to both religion and Math is much healthier for the mind and soul.

    If my reasoning above is incorrect – please correct me.

  2. The model in the link below shows that Kant was most certainly thinking along the correct lines. However his conclusion that existence ends as a positive is false. He came to this erroneous conclusion because he did not incorporate anti-matter into his analysis whereas, if he did, then he would have realised that the sum total of changes not only results in zero (absolute void – shunya or sunyata) but also a net neutrality (as opposed to a net positivity).

    I think the model explains itself in terms of the oppositions whether between +/- or +/+ or -/- and the centre of the model is the godhead/void/zero/neutrality.

    The last point is that the philosophical implications of a net positivity is the creation of a hierarchy from REALITY to realities. However, if net neutrality is the correct position then REALITY provides the home for the other realities in the form of a holarchy. The implications of this for conservative thought is obvious since no longer can the natural order be imagined as a hierarchy but has to be imagined instead as a holarchy.

    Similarly the metaphysical implications is that God is not a seperate REALITY of zero in relation to positive realities but that God is Everything.

  3. this is a very rigorous essay, and the comments are outstanding as well. much to ponder…but I’m not sure that my general perception of Kant has changed….

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