*This is the Appendix of A Reformulation of Dialectical Materialism*.

**A. Introduction**

Is all nature dialectical? That it is, was a fundamental premise of Engels in formulating dialectical materialism. In the preface to the second edition of **Anti-Dühring**, Engels writes that he had spent the best part of eight years in gaining knowledge of mathematics and natural science, for such knowledge “is essential to a conception of nature which is dialectical and at the same time materialist” [Ref. 16]. Thus Engels sought to understand the dialectical development of human society in terms of universal laws applicable to all aspects of nature:

*.”.. my recapitulation of mathematics and the natural sciences was undertaken in order to convince myself also in detail – of what in general I was not in doubt – that in nature, amid the welter of innumerable changes, the same dialectical laws of motion force their way through as those which in history govern the apparent fortuitousness of events; the same laws as those which similarly form the thread running through the history of the development of human thought and gradually rise to consciousness in the mind of man; the laws which Hegel first developed in all-embracing but mystic form, and which we made it one of our aims to strip of this mystic form and to bring clearly before the mind in their complete simplicity and universality.” [Ref. 17]*

Engels wrote at considerable length about dialectical development in two major works: **Anti-Dühring** (“AD”) and **Dialectics of Nature** [Ref. 18) (“DN”). In DN (p. 26) he formulated the three basic laws of dialectics, abstracting them from “the history of nature and human society”:

“*The law of the transformation of quantity into quality and* vice versa;

*The law of the interpenetration of opposites*;

*The law of the negation of the negation*.”

Engels considered these to be “the most general laws of these two aspects [nature and human society] of historical development, as well as of thought itself”, and he gave many examples and analyses of the manifestation of dialectical development in the physical sciences (physics, chemistry) and in mathematics, in AD and especially in DN.

We address in this paper the question of whether dialectics is applicable to the physical sciences and mathematics. Although a fundamental premise for Engels in formulating dialectical materialism, the applicability of dialectics to nature and mathematics has been essentially ignored by other classical elaborators of Marxism such as Marx, Lenin, Stalin, and Mao, who of course had other questions to occupy their time. Marx deferred to Engels’s expertise in such matters, and Stalin quoted Engels to establish the universality of dialectical development [Ref. 19]. Lenin made brief mention of dialectical contradiction in nature and mathematics:

“*In mathematics: + and -. Differential and integral*.

* In mechanics: action and reaction*.

*In physics: positive and negative electricity.*

*In chemistry: the combination and dissociation of atoms*.

*In social science: the class struggle*.” [Ref. 20]

And Mao quoted this passage to illustrate the universality of contradiction [Ref. 21]. There have been fairly recent attempts to establish the universality of dialectics, notably by the (now-defunct) Communist Labor Party, whose journal we shall quote later in connection with the application of dialectics to mathematics. But in addressing the question of whether dialectics is applicable to the physical sciences and mathematics, we must concentrate upon the two fundamental works of Engels, AD and DN.

The natural sciences have always been “bourgeois”, in the sense that they completely ignore Engels’s view that their subject matter exhibits dialectical development. There was, however, a major attempt made in the Soviet Union to reformulate the natural sciences in terms of dialectics, as pointed out by Holubnychy, summarizing the study of Joravsky [Ref. 22]:

“*Engels left materialistic dialectics as a more or less developed philosophy, but not a science or a formal scientific method. Yet, in the twenties, Russian Communist “believers” tried to force dialectical method on all sciences, including the natural sciences. The attempt failed dismally, and dialectics became seriously compromised as a result. Tacitly, the Russians concluded that it was of little practical value in education and that it should be replaced by the study of traditional logic*.” [ Ref. 23]

There is a very basic reason that this attempt to apply dialectics to the natural sciences fell flat on its face, so that Soviet scientists had to revert back to “bourgeois science”: nonliving nature is not dialectical. To establish this fact, we shall examine in detail Engels’s claim that the science of physics, as well as mathematics, exhibits dialectical behavior.

In examining Engels’s work, however, we are faced with a problem: if we deny his understanding of dialectics, then what can we say that dialectics is? If we don’t know what dialectics is, how can we say that physics and mathematics are not dialectical? Some definition of dialectics, independent of Engels’s own presentation of dialectics, is necessary in order to evaluate his claim, else we go around in circles. Let us therefore start with the assumption, which we consider to be valid, that Engels was basically right: the development of human society is dialectical, and dialectical materialism, utilizing the laws of dialectics, is indeed the science of society. What we need is the essence of dialectical development: it is irreversible, leading (by definition) to higher and higher forms, and it occurs through the development of internal contradictions. It is this understanding of dialectics which we shall use in evaluating Engels’s claim that physics and mathematics exhibit dialectical behavior.

There is one other point necessary to make, before commencing the evaluation of Engels’s work: I am a theoretical physicist, with the advanced academic training in physics and mathematics (for a Ph.D. degree) required for a successful research career. [I am now retired as a (Full) Professor of Medical Physics at Rush University in Chicago. My research, with some thirty peer-reviewed papers published in medical physics journals, has been concentrated on improving the radiation treatment of cancer using beams of electrons, photons (x-rays), and protons.] My training in physics and mathematics does not necessarily mean that my understanding of these fields is correct, but I am at least aware of their nuances, in a way that a person who only reads about them can never be. This advanced knowledge of physics and mathematics is the basis for my making such sweeping statements as that physics does not describe dialectical processes, or that mathematics contains no (logical) contradictions. Engels, for all his intellectual brilliance and contribution to human liberation, was so far as natural science goes always on the outside looking in, and this is why he was able to believe that dialectics, whose application to human society he so well understood, is also applicable to the physical sciences and mathematics.

**B. Physics: Change of State of Matter**

There are not very many ways in which one can construe the phenomena encompassed by the science of physics to behave dialectically. But there is one salient example, that of change of state of matter, which Guest presents in explaining Engels’s first law of dialectics, the transformation of quantity into quality and vice versa:

“*The simplest (and classical) example is the change of state [of] a substance, e.g. when a liquid becomes a gas (through boiling) or a solid (through freezing). Everyone knows that in such a case, gradual increase or decrease of temperature *produces* no departure from quality of liquidity until suddenly a point is reached where a complete transformation is effected. The liquid (as Hegel says) does not gradually become more and more gelatinous and semi-solid. It *leaps at one bound* from the liquid state to the solid*.” [Ref. 24, emphasis added]

Now Guest is mainly quoting Engels’s ideas here, and we shall have to see what Engels actually says on this question. But let us first evaluate this example from the standpoint of physics. What Guest is saying is simply wrong. Change of state of matter does not occur through change of temperature; change of temperature does not “produce” change of state at certain nodal points. Rather, the addition (or subtraction) of energy in the form of heat gradually changes the whole body from solid to liquid form (or vice versa); the body does not “leap at one bound from the liquid state to the solid'” What is actually going on, for example, in the melting of ice is that water molecules which had previously been bound closely together into a solid substance gain the energy (heat) to break these bonds and dissociate from each other. As more and more heat is added, more and more of these bonds are broken and, as a whole, the substance becomes a liquid. Throughout this process the temperature remains constant.

One must understand that the temperature of an object is an *effect*, not a *cause*, of physical processes. Knowing the temperature of objects enables one to know certain ways in which they may interact with each other. For example, if two isolated objects of the same temperature are brought into contact with each other, nothing will happen, but if they have different temperatures then the colder one will absorb (heat) energy from the hotter one, at a rate proportion to their (instantaneous) difference in temperature, until they have the same temperature (intermediate between their two previous temperatures). The physical process occurring here is the exchange of energy in the form of heat, and the temperatures of the two objects is but a *measure* of what is going on. To say that raising the temperature of ice past the melting point “produces” a change to the liquid state, is like saying that changing the speedometer reading of an automobile from 50 mph to 60 mph causes the automobile to go faster.

Engels studied physics at great length, and he was quite well aware of the physical process involved in change of state. In AD, pp. 79-81, he goes into considerable detail about the transfer of heat in the change of state, and is quite clear on its being a gradual process (in contrast to Guest, who seems to be more imbued with Hegel). But Engels is also quite clear about regarding change of state to be an example of the law of transformation of quantity into quality and vice versa:

“*This is precisely the Hegelian nodal line of measure relations, in which, at certain definite nodal points, the purely quantitative increase or decrease gives rise to a qualitative leap; for examp*le, *in the case of heated or cooled water, where boiling-point and freezing-point are the nodes at which – under normal pressure – the leap to a new state of aggregation takes place, and where consequently quantity is transformed into quality*.” [AD, p. 59]

And furthermore, in contradiction to his knowledge of the physical process involved, Engels states that “the merely quantitative change of temperature brings about a qualitative change in the condition of the water” (AD, p. 151, underlining added).

Thus, in this example, the problem is not simply Guest’s interpretation of Engels, it is Engels himself. But let us return to Guest, to the paragraph immediately preceding the one quoted above, to see what the stakes are:

“*This law is essential for an understanding of the rise of new qualities, and also for understanding the quantitative effects which may follow the appearance of such new qualities. It is one of the fundamental superiorities of dialectical over mechanical materialism that the former understands how new qualities can arise at certain points of quantitative change – points where the change in quantity literally becomes qualitative change*. “[Ref. 25]

Let us go beyond the content of the example which Guest then provides (about change of state of matter, which is simply wrong) to the form, and see whether this application might be useful if it happened to be physically valid. Is it true that “this law is essential for an understanding of the rise of new qualities”, in this case change of state of matter? The answer here has to be no!, expressed far more emphatically than would be proper here. Not only is dialectics not essential here, it could not possibly provide better understanding of change of state of matter, for this process is *completely* understood by “mechanical materialism”. And this example does not even attempt to provide *understanding* of the process of change of state: all dialectics does here is to make the observation that various forms of matter change state at particular temperatures, which is hardly a novel discovery. (This utter shallowness in the application of dialectics is reflected subsequently in the use of “Marxist-Leninist theory” by the international communist movement since the rise of Stalinism, as simply justification for whatever seems at the time to be appropriate to do.)

Change of state of matter is probably the clearest example which Engels provides of a dialectical process occurring in the field of physics, and let us examine it further to ascertain to what extent this process might be considered to be dialectical. Is this process, in terms of our own understanding of dialectical development, “irreversible, leading to higher and higher forms, and occurring through the development of internal contradictions”? Change of state of matter certainly doesn’t proceed through the development of internal contradictions; there is no self-movement here, but rather the (completely determined) reaction to external forces, causing an increase or decrease of energy within the object.

And how about the irreversibility of the process, the gaining by the object of higher and higher forms, as implied by the law of the negation of the negation? While change of state is certainly a qualitative change, it is also reversible by an opposite physical process: we can melt ice by exposing it to warmer air, but we can then freeze the water right back up by exposing it to colder air. If we want to call melting the ice its “negation”, then we would have to call the freezing process a “negation” (unless we were to give up all pretense of abstraction), so that the “negation of the negation” leaves us back where we started. Aside from being meaningless for practical purposes, such an interpretation of the law of the negation is hardly what we have in mind in our understanding of the development of human society – indeed, it is *because* we recognize that this development is dialectical that we would say that it is impossible for a feudal society, once having transformed itself to capitalism, to revert back to feudalism.

**C. Physics: Other Examples**

In attempting to apply dialectics to the physical sciences, Engels and his successors formulate (incipient) theories in terms of interpenetrating opposites. At best, such formulations are trivial, not furthering our understanding of the processes involved but at least not hurting our understanding (unless one decides to stop there, thinking that one actually knows something). For example, in Section A we have already seen the examples provided by Lenin5: action and reaction in mechanics (a branch of physics), positive and negative electricity in physics, the combination and dissociation of atoms in chemistry. The problem here isn’t the recognition of dualities, such as two kinds of electric charge – it is their treatment as interpenetrating opposites, a mystical conception which leads us nowhere in understanding (nonliving) physical reality. In order to gain quantitative understanding of reality, (physical) scientists have had to develop their fields in terms of mechanical cause-and-effect relationships rather than dialectical mysticism, and the incredible success of “mechanical materialism” in developing the physical sciences is the ultimate proof of the inapplicability of dialectics to these fields.

But let us examine Engels’s formulations when they are not trivial. In that case, because Engels doesn’t really understand physics, and because he tries to use dialectics to overcome this lack of understanding (*i.e*., to achieve qualitative understanding because he has not the option of achieving quantitative understanding), what Engels says is often muddled, and occasionally dead wrong. For example, Engels (DN, pp. 46-47) doesn’t understand the difference between force and energy, and he tries to place them in terms of polar opposites, attraction and repulsion. In pp. 52-55, in trying to disprove a theory of the contemporary physicist Helmholtz, Engels decides that heat is a “repulsive force”, acting in contradiction to the attraction of gravity and chemical forces. But physics is a quantitative science, and one cannot just play around with words (qualitative formulations) as Engels does; Engels is hopelessly lost in addressing Helmholtz’s theory. Engels considers Helmholtz to be confused about the notion of force, and considers such confusion to be “the best proof that it [the notion of force] is in general not susceptible of scientific use in all branches of investigation which go beyond the calculations of mechanics” (p. 55), in particular physics and chemistry. Here Engels is not only centuries behind his time, but, because of his reliance on dialectics to understand that which can be understood only through mechanical materialism, he would throw out the whole science of physics. Since Newton’s formulation of the laws of motion of matter, the notion of force has been integral to physics.

Engels makes other erroneous conclusions in DN. He doesn’t understand the difference between momentum and energy, and wrestles with his confusion at considerable length (pp. 64-71). He states a “law of the indestructibility and uncreatibility of motion” (apparently corresponding to the law of conservation of energy) as expressing that “the sum of all attractions in the universe is equal to the sum of all repulsions” (p. 38), which is nonsensical, and goes on to formulate the (incorrect) thesis that bodies at very great distances from each other repel one another, instead of experiencing gravitational attraction (p. 259). He confuses entropy and energy, and in view of the (correct) law of conservation of energy, he dismisses (pp. 205, 216) the (correct) law of thermodynamics which states that the entropy of the universe is continually increasing.

Engels’s understanding of motion of matter is outright mysticism, starting with his statement that “motion is the mode of existence of matter” (AD, p. 75). Such a conception has nothing to do with Newton’s laws of motion, upon which the science of physics rests, nor could it be incorporated into a *quantitative* theory of physical processes. In Engels’s view,

“*Motion itself is a contradiction: even simple mechanical change of position can only come about through a body being at one and the same time both in one place and in another place, being in one and the same place and also not in it. And the continuous origination and simultaneous solution of this contradiction is precisely what motion is*.” [AD, pp. 144-145]

This is gibberish! Engels’s view that a body can be in two places at once would obliterate the science of physics, just as his view that a body can at the same instant both be in a place and not be there is objectively an attack on human rationality. Engels actually has in mind the notion of qualitative change, and it is quite possible (and indeed necessary) to understand qualitative change of human society in the contradictory way which he poses, although certainly not as a logical contradiction. But we are here dealing with mechanical motion, which most assuredly is not a manifestation of qualitative change.

Coming back to our understanding of dialectical development, as being irreversible and leading to higher and higher forms, we see that there are examples from physics which might possibly be construed as exhibiting such motion. As examples of irreversible processes, we have that the flow of heat energy is always from a hotter to a colder body, and that the entropy of the universe as a whole is continually increasing. Entropy can be considered to be lack of order, so the universe as a whole is undergoing qualitative change, moving to “higher and higher levels” of disorder. A dialectical process defines the meaning for itself of “higher and higher levels”, but increasing disorder would hardly seem to be what we mean by progressive dialectical development. Another example of irreversible qualitative development is the cycle of stars, from their birth to their death, but such development is completely preordained, like the motion of a clock once it is wound up; it is hardly useful to try to understand the cycle of stars in dialectical terms, for their development is well understood through the laws of physics (*i.e.*, through mechanical materialism).

And this is the problem with imposing dialectics on the science of physics: such imposition is completely mechanical and arbitrary, and does not aid our understanding of physical phenomena. In the field of chemistry, Engels gives (DN, p. 229), as an example of the law of transformation of quantity into quality, molecular oxygen (two atoms of atomic oxygen combined into a molecule) and ozone (three atoms of atomic oxygen). Well, it’s true that molecular oxygen and ozone have markedly different chemical properties, but so what? How does thinking in terms of “the law of transformation of quantity into quality” help one to understand the difference in their chemical properties? Once again Engels is on the outside looking in, and his application of dialectics to the physical sciences only serves to trivialize dialectics and hinder us from understanding real dialectical processes. Physics in fact does not exhibit dialectical processes in any meaningful sense, and no competent physicist would attempt to impose dialectics on this science.

**D. Mathematics**

Let us be clear at the outset: mathematics is not a science. A science has as its subject matter a body of (natural) phenomena which can be understood in terms of an appropriate set of general laws, *i.e*., a body of “qualitatively similar” phenomena. On the other hand, mathematics deals with logical systems each developed from a set of (unquestionable) axioms. It *may* happen that a particular set of axioms is such as to lead to a logical system which can successfully be used as an “analytic superstructure” to a particular science, for example the real number system with the usual arithmetic operations, or the system of calculus used in physics. Indeed, mathematicians have spent the great bulk of their time developing mathematical systems applicable to the sciences, for this has made their work practical to society (and therefore supportable by society); furthermore, natural phenomena have provided mathematicians with well-defined, meaningful problems to tackle. But in spite of the very close relationship between the sciences and mathematics, the two are of completely different character: a science is necessarily *materialist*, requiring continual experimentation and practical observation to verify its body of theory (which in turn is a summation of observed phenomena); but a mathematical system is *idealist*, with the validity of its propositions determined entirely by whether they flow logically from the (given) axioms of the system.

Thus, a mathematical system contains no (logical) contradictions – if the mathematical system you’ve developed from a particular set of axioms does contain contradictions, you’ve got to start over and do it right this time! (And if your mathematical system isn’t applicable to a particular science, then you have to start again from scratch, with a new set of axioms.) With this understanding of mathematics, we see immediately that dialectics has no application to mathematics. Dialectical development occurs through the development of internal contradictions, but there is no self-movement in mathematics, and indeed there are no contradictions, internal or not.

Engels attempted to foist dialectical concepts onto mathematics, but it is clear from the passages on mathematics in AD and DN that he had essentially no grasp of the subject. His comments on algebra are painfully trivial and naïve, such as pp. 198-200 and 251-255 of DN, as well as:

“*But even lower mathematics teems with contradictions. It is for example a contradiction that a root of A should be a power of A, and yet A½= the square root of A. It is a contradiction that a negative quantity should be the square of anything, for every negative quantity multiplied by itself gives a positive square. The square root of minus one is therefore not only a contradiction, but even an absurd contradiction, a real absurdity. And yet the square root of ‑1 is in many cases a necessary result of correct mathematical operations. Furthermore, where would mathematics – lower or higher – be, if it were prohibited from operating with the square root of ‑1?*” [AD, p. 146]

To illustrate the law of the negation of the negation, Engels gives the following example:

*Let us take any algebraic quantity whatever: for example, a. If this is negated, we get ‑a (minus a). If we negate that negation, by multiplying ‑a by ‑a, we get +a squaed, i.e. the original positive quantity, but at a higher degree, raised to its second power. In this case also it makes no difference that we can obtain the same a squared by multiplying the positive a by itself, thus likewise getting a squared. For the negated negation is so securely entrenched in a squared that the latter always has two square roots, namely a and ‑a. And the fact that it is impossible to get rid of the negated negation, the negative root of the square, acquires very obvious significance as soon as we come to quadratic equations*.” [AD, p. 164]

But Engels is just playing with words here, making no contribution toward understanding how to use algebra or toward understanding the nature of algebra, and in fact mystifying algebraic processes.

Regarding calculus, Engels fares even worse. Part of the problem seems to be his reliance on a book then eighty years old (AD, p. 470, footnotes 197 and 198), written some fifty years before calculus was placed on a rigorous footing with the introduction of the concept of limit by Cauchy. This study of Bossut’s outdated book appears to have left Engels with his erroneous ideas on calculus, such as that curves and straight lines can be equated to one another (AD, p. 144; DN, pp. 200-201), and that 0/0 has mathematical meaning (AD, pp. 164-165 and 412-413). In fact, Engels points out that 0/0 represents a contradiction, but his response, based on his philosophical outlook, is acceptance:

“*Therefore, dy/dx, the ratio between the differentials of x and y, is equal to 0/0, but 0/0 taken as the expression of y/x. I only mention in passing that this ratio between two quantities which have disappeared, caught at the moment of their disappearance, is a contradiction; however, it cannot disturb us any more than it has disturbed the whole of mathematics for almost two hundred years*.” [AD, pp. 164-165]

In contrast, it did disturb the great mathematician Cauchy, and he rooted out such contradictions from calculus, thereby making it possible to develop it into an advanced instrument for understanding nature.

Mathematicians have of course ignored Engels’s efforts to foist dialectics onto their field, but we are fortunate in being able to examine a contemporary attempt to do so, albeit only at the level of algebra. In May of 1978, the Communist Labor Party held a philosophy seminar, and one of the conference papers which it subsequently published was devoted to dialectics in mathematics. We conclude this section by quoting a paragraph explaining “dialectical movement in equation-solving”, to illustrate how mechanical and vacuous such attempts can be:

“*The equation 3X+4=19 poses the contradiction “What is X?”. In elementary school, children memorize the mathematical law: “When equals are added to equals the results are equal”, an expression of dialectical interconnectedness of any equation. The movement which resolves the given contradiction begins by applying this law and the equation ‑4=-4 to obtain 3X+4+(‑4)= 19+(‑4), or 3X=15. The oppositeness of addition and subtraction has been utilized to negate the 4, allowing it to move from one pole of the equation and penetrate the other, thereby changing both poles. A mutual penetration. We now use a version of the above law with “added to” replaced with “multiplied by” and the equation 1/3=1/3 to obtain 1/3x3X=1/3×15 or X=5. This time the oppositeness of multiplication and division is used to negate the 3 and again mutual penetration takes place resulting in the solution of the equation*“.[ Ref 26]

**E. Conclusion**

It is most interesting to read what the Soviets had to say about dialectics two decades after the Bolshevik Revolution, and we can refer to the **Textbook of Marxist Philosophy**, prepared by the Leningrad Institute of Philosophy under the direction of M. Shirokov [Ref 27], in 1937. This work is a lengthy, very comprehensive review of Marxist and pre-Marxist philosophy, with particular application to the political struggles and questions confronting the Soviet state at the time. As was usual during the Stalin era and since, it reduces Marxist theory to justification, after the fact, of the political decisions taken, and one finds in it, for example, “proof” that Stalin’s political opponents were wrong philosophically, and that classes and class struggle are immediately to disappear under socialism. But it was also written after an unsuccessful attempt had been made to reformulate the natural sciences in terms of dialectics, as mentioned earlier [Ref 16, p. 67], and what is most revealing is what the work does not say. The application of dialectics to mathematics is simply ignored, and, while lip service is given to Engels’s views that dialectics is applicable to physics and chemistry, Shirokov has stepped very gingerly to avoid repeating Engels’s obvious errors. Indeed, there appear to be only two clear-cut false statements in this area: that “a moving point” is simultaneously found and not found in a given spot” (p. 152), and that the development of physics and chemistry demanded a new methodological system, beyond mechanical materialism (p. 234). On the one hand it is good that Shirokov largely avoided Engels’s erroneous views on the application of dialectics to the physical sciences and mathematics, but on the other hand it is most unfortunate that he could not openly identify Engels’s errors, for by that time in the Soviet Union Marxist-Leninist theory had been transformed into a state religion in the service of a new exploitative class.

We trust we have demonstrated that the physical sciences and mathematics do not exhibit dialectical development, in the sense which we understand it: irreversible development, leading to higher and higher forms, and occurring through the development of internal contradictions. Thus we must forego Engels’s view, so comforting and yet so wrong, that the dialectical development of human society is but one manifestation of the universal laws guiding all natural processes. Comforting indeed this view is, for we are assured thereby that the monstrous system of capitalism, like every other natural phenomenon, will necessarily come to pass, preordained to be succeeded by the higher form of communism – we are not idealists looking for Utopia, but have grasped the essence of all natural development!

So we cannot invoke universality of dialectical development. So what? The fact remains that human society, as well as all life processes, exhibits dialectical development in the sense understood by Marx and Engels, by Lenin and Mao. We do not need external justification for our observation that society develops dialectically; what we need is detailed understanding of this dialectical development. And such understanding, the raising of dialectical materialism (“the science of society”) from an empirical science to a theoretical science, requires the systematization of dialectics, which is impossible to carry out so long as we remain muddled in thinking that dialectics is applicable to nonliving phenomena. Thus we must disabuse ourselves, once and for all, of the notion that all nature is dialectical.

**REFERENCES**

16. Frederick Engels, **Anti-Dühring** (Progress Publishers, Moscow, 1969), p. 15.

17. *Ibid.*, p. 16.

18. Frederick Engels, **Dialectics of Nature** (International Publishers, New York, 1940).

19. Joseph Stalin, **Dialectical and Historical Materialism** (International Publishers, New York, 1940), pp. 9-11.

20. V.I. Lenin, “On the Question of Dialectics”. In **Collected Works**, Vol. 38 (Moscow, 1958), p. 357.

21. Mao Tsetung, “On Contradiction” (August 1937). In **Four Essays on Philosophy** (Foreign Languages Press, Beijing, 1968), p. 32.

22. David Joravsky, **Soviet Marxism and Natural Science, 1917-1932** (Columbia University Press, New York, 1961).

23. Vsevolod Holubnychy, “Mao Tse-tung’s Materialistic Dialectics”, *China Quarterly* **19**, pp. 3-37 (July-September 1964).

24. David Guest, **Lectures on Marxist Philosophy** (New Book Centre, Calcutta, 1963), p. 39. (First published in 1939 under the title **A Textbook of Dialectical Materialism**.)

25. *Ibid.*, p. 39.

26. Mike B., “Dialectics in Mathematics”, *Proletariat* **4**, No. 4, pp. 33-37 (Winter 1978).

27. M. Shirokov, **Textbook of Marxist Philosophy**, 1937. Prepared by the Leningrad Institute of Philosophy under the direct of M. Shirokov, and reprinted by Proletarian Publishers, P.O. Box 40273, San Francisco 94140.